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in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said .

do we know why someone wrote the rosseta stone ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

do egyptians still celebrate the ancient gods today or just 1 god ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids .

how many kings were mummified ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids .

what were the mummified kings names ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great .

how did the kings rule ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great .

was egypt not a monarchy during the intermediate period ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history .

what happened to the pharaohs of the old kingdom that it meant the end of the old kingdom ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages .

did the pyramids have traps-pits , loose steps , etc- to prevent anyone from invading the tombs ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great .

2 could i please have a quick recap of what the `` neolithic period '' was ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here .

during the egyptian times , was anyone allowed to be buried in the tombs , or under the pyramids ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization .

are aten and ra the same person ( the sun god ) ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce .

is there gold inside pyramids and are the king inside dead in the pyramid ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile .

when was upper and and lower egypt united ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages .

what light source was used in underground tombs and passages ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here .

can daylight be reflected by polished brass mirrors with enough intensity to be practical over the distances required ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization .

isent houres the god of the sky ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings .

what were the pharaohs referred as before the new kindom ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper .

is the nile river the only river that flows from south to north ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea .

and , why do most rivers in the world flow from north to south ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

how was life different in each part of egypt ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

how many pyramids exist in egypt approximately ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story .

if pyramids were trapped , who disarmed them to place the pharaoh ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important .

is n't amun ra / ra the sun god ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids .

how many kings were mummified ?

in this video , we are going to give ourselves an overview of ancient egypt , which corresponds geographically pretty closely to the modern day state of egypt in northeast africa . now the central feature in both ancient egypt and in modern egypt is the nile river that you see in blue right over here . and the nile river is one of the great rivers of the world . it rivals the amazon river as the longest river and it sources the tributaries of the nile rover start even south of this picture and the water flows northward and eventually its delta reaches the mediterranean sea . the delta , which is where a river opens into the sea , is called a delta because , as you can see , these rivers , you can even see it from the satellite pictures right over here , they start branching up a bunch and you have this upside down triangular region , which looks a little bit like an upside down greek letter delta , so that 's why river delta is called that . and this one just happens to be upside down . if it was flowing the other way , it would be a right-side-up delta . so the nile river , it flows from , you could say , eastern mid-africa up into the mediterranean sea and because it has this northward flow , the southern parts of the river are upriver and they are actually called the upper nile . so , upper . the upper nile is actually south of the lower nile , of the lower nile . and once again , that 's because the upper nile is up river , it 's also flowing from higher elevations to lower elevations . so as you go south , you get to higher and higher elevations . now , the reason why the river is so important , we studied this multiple times , rivers are a source of fresh water , when they flood they make the surrounding soil fertile , they 're suitable for agriculture , and the nile valley is one of the first places that we see agriculture emerging during the neolithic period . in fact , human settlement we believe was along this nile river valley as far as 6,000 bce or 8,000 years ago , and it might have been there even further back in time . and because you have that agriculture , it allowed for higher population densities , which allowed for more specialization of labor and more complex societies . it 's not a coincidence that some of the first , that one of the first great civilizations emerged here . now , the story of the nile river , or of egypt , and actually they are tied very closely , even though egypt is considered a lot of this region , most of the human population , this is true even today , is right along the river , around that fertile soil , where the agriculture actually occurs . in fact , this was so important to the ancient egyptians that their whole calendar , their seasons , were based on what the nile river was doing . they had a season called the inundation , or the flooding of the river , which makes the soil fertile . they had a season of growth , which is now talking about the growth of the crops and they had a season of harvest . and so you had people in this valley for thousands of years , but when we talk about ancient egypt , we formally talk about it as a civilization around 3,100 , 3,150 bce . and this is where we get to our timeline right over here . so we 're talking about right around there on our timeline and the reason why this is considered the beginning of the ancient egyptian civilization is this is when we believe that upper and lower egypt were first united under the king and there 's different names used , narmer sometimes or menes . i 'm going to mispronounce things every now and then and i 'm probably doing it here as well . and so he was the king that unified upper and lower egypt into an empire and the empire , as we will see , which lasted thousands of years , every one of these spaces is a hundred years . we 're gon na go over huge time span , but the ancient egyptian civilization is roughly divided into three kingdoms . you have the old kingdom , which went from about , right from about the 27th century bce up to about the 17th century bce . you have the middle kingdom and you have the new kingdom . and once again , this is spanning right over here over a thousand years of history . and in between those , you have these intermediate periods where the kingdom or the empire was a little bit more fragmented . you have in some of these intermediate periods , you have some foreign rule . but just to get a sense of some of what happened over this thousands of years , and i 'm kind of laughing in my head because it 's hard to cover over two , 3,000 years , in the course of just a few minutes , but this will give you a sense of what ancient egyptian civilization was all about . now the kings are referred to as pharaohs but as we 'll see that term pharaoh is not really used until we get to the new kingdom . but i will refer to the kings as pharaohs throughout this video , just to say , hey these are the egyptian kings . and the old kingdom is probably most known today in our popular culture for what we most associate with ancient egypt and that is the pyramids . and here , right over here are the pyramids , there 's the great pyramid of giza , which is near modern-day cairo today . this is the sphinx and they were built in that old period under the pharaohs sneferu and khufu , right over here in the 26th century bce . and we are still trying to get a better understanding of how this was done . we actually now do n't believe that it was done by slave labor , but instead it was done during , you could say , the off season by the peasants as a form of taxation . okay , you 're done planting or harvesting your crops ? well now that you have some time , and this shows actually the importance of agriculture for freeing people up , so to speak , why do n't you help the pharaohs built these massive tombs , which i 've seen various estimates that it might have taken some place between 10 and 100,000 people several decades to build each . but these are even today , these were built over 4,500 years ago , are some of the most iconic symbols that humanity has ever created . and the reason why we know so much about ancient egypt is that we have been able to decipher their writing . it 's a symbolic , they have these pictographs , these hieroglyphics , i 'm sure you 've heard of the word before , and for a while we had no idea what they said . we would see these encryptions in these tombs and we had a sense that , okay these tombs , especially things like the pyramids would be for these great kings , we could tell that it was a stratified society , that nobility had better tombs than others , but we did n't really have a good sense of what was going on until we discovered this , which is the rosetta stone , which was discovered in 1799 . the reason why this is so valuable is it has the same text written in three different languages . it has it written in the hieroglyphs of the ancient egyptians , and it has it written in a later script used in egypt , called demotic egyptian , and most importantly , it has it also written in greek . and so historians were able to say , okay , we can now start to decipher what these symbols mean because we have a translation of them and that 's why it 's one of the first civilizations where we 're able to put the picture together . and hieroglyphics are one of the first forms of writing . but let 's now go on in our journey through thousands of years of ancient egyptian civilization . between the old kingdom and the middle kingdom , you have the first intermediate period and then you have the middle kingdom and then you have the hyksos , which are semitic people , semitic referring to their language being of the same family as semitic languages like arabic , or hebrew , or aramaic . but then you have the new kingdom , and the new kingdom is considered to be the peak of ancient egypt . it 's really the height of their technology , it 's the height of their military capability . and there are several pharaohs that are worthy of note in the new kingdom . the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection . you have the god amun here and his first name amenhotep , it means amun is satisfied . what is considered kind of the equivalent of zeus , you have the god here horus , once again a very significant god at different times in egypt , but what was interesting about amenhotep the fourth or akhenaton , whichever name you want to use , is he decided , no , no , no , i do n't like this pantheon , this polytheistic religion that we have , i wan na worship one god , and the god that he decides to worship is really the , you could consider it the sun god , or the sun disc , and its representation looks something like this and it was referred to as aten and so he changes his name to akhenaton and he actually starts to try to get rid of evidence of these other gods or to make them a lot less important . and so the reason why that 's notable is this is viewed as perhaps one of the first attempts at monotheism , at least within this ancient egyptian civilization . he 's also noted for giving a lot of power to his wife , to the queen , nefertiti , who some people say was second in command , or even co-ruled alongside him . now he was also famous because after his death , eventually , his son , king tut , tutankhamen , comes to power . and the reason why king tut , as he 's often known , although it 's tutankhamen , is known is because we were able to find his tombs in relatively good order and so he 's become a popular part of the imagination . and he 's known as a child pharaoh . he comes to power when he 's very young , he dies at 18 and so it 's kind of an interesting story . now , most prominent amongst all of the pharaohs across egyptian history , and this is also in the new kingdom , comes a little bit after tutankhamen , is ramses the second . and ramses the second , who emerges here in the 13th century , and he rules for most of the 13th century bce , he represents really the peak of egypt , ancient egypt , as a military power . he 's famous for the battle at kaddish , which is the earliest battle where we actually know what the tactics and the formations were and it was with the also significant hittite empire in 1274 bce , this is an image drawn much , much later , of the battle of kaddish . the battle , we now believe , might have been a bit of a stalemate , ramses the second was n't able to capture kaddish , but has told us a lot about military tactics and strategy and formation of that time . historians today think it might have been the largest chariot battle maybe ever . so this was a significant thing that happened . now , eventually the new kingdom does collapse , as we get to the end of the second millennium , and then over the next several hundreds of years , we 're talking about a very long period of time , it gets fragmented , you have several rulers , you have the kushites rule from the upper nile , the kushites were in this area right over here . they rule for a brief period . the assyrians , that 's a mesopotamian civilization , they rule for a small period of time , and then eventually and we talk about this in some detail in other videos , you have the persians take over , you have cambyses , cyrus the great 's son , he 's able to rule over , he 's able to conquer egypt and egypt becomes part of the achaemenid empire for a while until the conquering of alexander the great . and after alexander the great dies , one of his generals and his dynasty takes over , ptolemaic egypt and now it 's being ruled by foreigners , well it 's been ruled by foreigners for a while now , but now it 's by the greeks and the famous cleopatra , who 's considered a pharaoh of egypt , she 's actually greek by blood , she 's actually the one that seduced you could say julius cesar and marc antony and after cleopatra 's death , more and more , actually eventually it becomes part of rome . so as you can see we covered this enormous large time period in history , one of the most significant civilizations in all of history , one of the most famous poems about civilizations and rulers , about ramses the second , the poem ozymandias was named after him . you have some of the great cities of the ancient world , thebes , which was the capital during parts of the new kingdom and the middle kingdom , you have memphis , which was one of the , some people say founded by menes and the capital of the old kingdom . these were all happening in ancient egypt .

the first is , he was born amenhotep or he was originally known as amenhotep the fourth and then he eventually names himself akhenaton and akhenaton means effective for aton , aton being a significant egyptian god . and the reason why he changed his name is he decides that , okay we have , the egyptians have this huge pantheon of gods . here is just the some of them right over here , this is the god osiris , often associated with the afterlife or transition , regeneration , resurrection .

were there any gods before ra and all the other gods came ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels .

can you regress to lower level even when those needs have previously been fulfilled ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

do you have to climb the hierarchy sequentially ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest .

can a person ever reach the top of maslow 's pyramid ; in other words , can a person ever reach their full potential ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well .

can we have love without safety.. ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid .

did maslow make one more shape for our basic needs ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid .

is there one more person like maslow in the world ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

is the maslow 's hierarchy of needs still used by psychologist ?

so we talked about before that there 's five approaches in understanding motivation . and one of these approaches is called maslow 's hierarchy of needs . and it 's actually broken down into a pyramid . so it looks just like this . and it was created by famous psychologist named maslow . so maslow said that we have needs that need to be fulfilled in a specific order . and it has to start from the bottom of the pyramid all the way to the top . so our most basic need is our physiological need . so this can include anything from food , water , breathing , sleep . all of these are essential needs to survive , basically . the second level is our need for safety , so safety of resources , safety of employment , safety in our health , property . so all of these are basic needs as well . but they can only be fulfilled when our physiological needs are fulfilled . so we call these two levels the basic levels . now , he went on to name a third level , and this is our level of love , our need for love , our need to belong , our need to have friends and family . so this level of needs is what we call our social needs . the fourth level is our need for esteem , self-esteem . so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level . and the last level is called self-actualization . it 's a big word , but it 's basically our need for wanting morality , a sense of morality , a need for acceptance and also creativity . so we call this our full potential . so think of this as climbing mount everest . you have to start at the bottom . but then , along the way , you 're going to have different checkpoints . each of these checkpoints are managed by all the sherpas on the mountain . you ca n't go from the bottom to the next level unless you check in with the sherpa , and he makes sure that you 're ok , you 've eaten properly , you 're getting enough rest , and only then can you jump to the next level . again , a sherpa there at the higher level is going to check and make sure you 're breathing ok , you 're getting enough oxygen , and so on . so you get to the next checkpoint and the next checkpoint , and finally , you 're at the top , where you 've realized your maximum potential . so this is called maslow 's hierarchy of human needs .

so we like to feel confident and have a sense of achievement in what we do . so this level is called our level of respect . we like to gain respect from others when we reach this level .

or being porn into an ethiopian poor family , am i stuck at level 1 or 2 since most ethiopians die by the age of 40 ?

( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studies the literal [ br ] meaning of the words or sentences that we use . but very often , we communicate more than the literal meanings [ br ] of the words we use . and this is one of the main things that pragmatics studies . for example , if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . but i did n't actually say that . i just stated two facts : one , `` i 'm going to montreal this week , '' and second , `` my mother lives in montreal . '' but everybody 's gon na naturally understand that those facts are connected . that in fact going to see my mother is my reason for going to montreal . here 's another example . suppose somebody asks me `` are you coming to the party on friday ? '' and i say `` i have to work . '' they 're gon na understand that [ br ] i ca n't make it to the party because i have to work . but again , i did n't say that . it could be that i have to [ br ] work earlier in the day . all i said was that i [ br ] have to work on friday . i did n't say anything about [ br ] it conflicting with the party . but you 're naturally gon na [ br ] understand my reply of `` i have to work on [ br ] friday '' as being a reason for me not coming to the party . in some cases , we can even [ br ] use the very same words in different situations and communicate completely [ br ] different things . for example , if a teacher [ br ] writes on a report card for a first grade student [ br ] '' bob has wonderful penmanship , '' it 's gon na communicate just that : bob is doing very well [ br ] at handwriting in class . that 's something you wan na [ br ] master in the first grade . on the other hand , suppose [ br ] a professor 's writing a letter of recommendation for one of her philosophy students applying for a prestigious award in philosophy . and all she writes in the letter is [ br ] '' bob has wonderful penmanship . '' well , in addition to [ br ] communicating something about bob 's penmanship , [ br ] that 's gon na communicate that the professor does n't think bob is a very good philosopher [ br ] or deserving of the award . herbert paul grice , a philosopher who lived from 1913 to 1988 , was the first who tried [ br ] to explain this phenomenon in his paper `` logic and conversation . '' and much of what he [ br ] said laid the foundation for the study of pragmatics today . grice invented the term `` implicature . '' an implicature is whatever is meant , but not literally said . things that are suggested , [ br ] implied , or hinted at . one very important kind of implicature that he talked about is [ br ] conversational implicature : implicatures that come about due to general features of conversation . and remember here again , [ br ] when we 're talking about conversations , we 're often [ br ] including things like books , letter-writing and so on . what grice observed is that , in general , conversations are cooperative efforts . people aim to understand [ br ] each other and be understood . they wan na give and receive information . they wan na influence each [ br ] other and be influenced . people in conversation [ br ] generally do n't say just a bunch of disconnected remarks . and even in the most [ br ] casual of conversations , there 's generally some sort of goal or purpose to the conversation . we do n't just idly say random things for no reason at all . grice took all these [ br ] observations and proposed that what we 're doing is sort [ br ] of tacitly following these rational rules of conversation . rational rules are [ br ] rules that people follow because we 're rational creatures , as opposed to , say , conventional rules . so for example , some countries drive on the left side of the [ br ] road and some countries , like ours , drive on the right . that 's just a conventional rule . one is not better than the other . but the fact that we [ br ] follow such rules at all , that we do n't just drive [ br ] in any direction we want , that 's a rational rule . that 's our way of [ br ] cooperating with each other . that 's how we get to where we 're going all in one piece , as opposed [ br ] to crashing into each other . grice summarized these observations , this idea that conversations [ br ] are cooperative activities among rational agents , [ br ] with his central rule that he called the [ br ] '' cooperative principle . '' the cooperative principle is `` make your conversational contribution such as is required , at the [ br ] stage at which it occurs , by the accepted purpose or [ br ] direction of the talk exchange in which you are engaged . grice further explained [ br ] the cooperative principle by giving four rules , or maxims , that people generally [ br ] follow in conversation . recall that these rules , or maxims , are not supposed to be conventional rules that we happen to follow [ br ] in conversations , but rules that govern rational [ br ] cooperative activity in general . the first is the maxim of quantity : make your contribution as [ br ] informative as is required for the current purposes of the exchange , and do not make your contribution more informative than is required . for example , if we 're talking about going to see a movie tonight and we 're trying to [ br ] decide what movie to see , and you ask me `` okay , what 's playing ? `` , following the first part [ br ] of the maxim of quantity , i should make my contribution [ br ] as informative as is required . and this just seems perfectly rational . i 'm going to read you , [ br ] say , all the times of the movies at our [ br ] local theater tonight , as opposed to just reading you the time of one movie that 's playing . but i 'm also gon na follow [ br ] the second maxim of quantity , `` do not make your [ br ] contribution more informative `` than is required . '' i 'm not , for example , [ br ] gon na tell you the times of all the movies playing [ br ] across the entire country . even though in some sense it 's an answer to your question of `` what [ br ] movies are playing tonight ? '' it 's more informative than is required by the conversational purpose . in the same way , this is [ br ] the sort of rule we follow in lots of cooperative activities . so for example , if we 're [ br ] fixing a car together and you need four screws [ br ] to screw in the next piece , i 'm not gon na hand you two [ br ] screws , which is not enough , or six screws , which is too many . the next maxim is the maxim of quality , which is simply `` try to [ br ] make your contribution `` one that is true , `` do not say what you believe to be false , `` and do not say that for which [ br ] you lack adequate evidence . '' so , back to the movie example , if we 're talking about going [ br ] to see a movie tonight and we 're trying to decide what to see and you ask me what 's playing , i should n't tell you something [ br ] that i know is not playing , something i know to be false . or i should n't tell you [ br ] a certain movie 's playing at a certain time if i [ br ] have n't looked it up yet , if i do n't have adequate evidence . again , this is a rule that we follow in regular cooperative activity [ br ] outside of conversation . it 's just something that rational people , when they 're trying to cooperate [ br ] with each other , will do . for example , if we 're trying [ br ] to bake a cake together and it 's time to add the sugar , i should n't give you the salt , [ br ] pretending that it 's sugar . the next maxim is the maxim of relation , which is simply `` be relevant : `` say things that are relevant , `` do n't say things that are irrelevant . '' so again , if we 're talking about going to see a movie tonight , i [ br ] should n't start telling you about a good book that i read , unless of course i signal [ br ] a change in conversation . similarly , if we 're trying to bake a cake and it 's time to add the sugar , i should n't pass you a [ br ] book , or even an oven mitt , even though the oven mitt may be relevant at a later point in the cake-making . the final maxim has to do not with the content of what we [ br ] say , but with the way in which you say it , and that 's the maxim of manner . more specifically , grice says [ br ] '' we should be perspicuous : `` avoid obscurity of [ br ] expression , avoid ambiguity , `` be brief , and be orderly . '' so , for example , if we 're talking again about going to see a movie tonight and a particular movie 's [ br ] playing at the local theater , i should just say so , [ br ] namely something like `` movie x is playing at our local theater . '' and not something like `` our local theater will be displaying a [ br ] series of still images on a reel tonight . '' if i say something using an [ br ] odd expression like that , you 're gon na start thinking i 'm trying to communicate something else , something like `` i do n't [ br ] think a real movie 's playing at the theater tonight . '' similarly , if we 're [ br ] baking a cake together and it 's time to add the sugar , i should just directly hand you the sugar . i should n't , for example , [ br ] draw you a treasure map to the sugar . these maxims , together with [ br ] the cooperative principle , explain how people communicate more than the literal meanings of [ br ] their words in conversation . people assume tacitly [ br ] that the speaker 's obeying all the maxims she can and [ br ] the cooperative principle . these facts , plus facts about the purpose of the conversation , the [ br ] context of the conversation , and the general things [ br ] we know about the world , our world knowledge , allow us to calculate these implicatures , these [ br ] things that are hinted at , suggested , or otherwise communicated over and above the literal [ br ] meanings of our words . so let 's return to the [ br ] examples from the beginning and see how grice will [ br ] explain how those things get communicated . recall that i said that if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . and now we can understand why . you assume i 'm being a cooperative conversational participant , [ br ] and only saying things that , for example , are [ br ] relevant to each other . i 'm not giving you a bunch [ br ] of disconnected remarks that are facts about the world . well , the only way that the fact that my mother lives in montreal is relevant to my going to montreal is that i 'm going to [ br ] montreal to visit my mother . there 's a similar explanation [ br ] for the next example . so again , suppose somebody asks me `` are you coming to the party on friday ? '' and i answer `` i have to work , '' and you understand that [ br ] i 'm not going to the party because i have to work . well again , you 're gon na assume that i 'm observing the [ br ] cooperative principle and all the maxims . and again , i 'd be infringing [ br ] on the maxim `` be relevant '' if my statement `` i have to work '' was n't somehow an answer to the question `` are you going to the party on friday ? '' i would also be infringing [ br ] on the maxim of quantity , `` be as informative as required , '' since i would n't have provided [ br ] an answer to the question if `` i have to work '' was n't [ br ] an answer to the question `` are you going to the party on friday ? '' and finally , we can employ a little bit of our world knowledge [ br ] there because we know that if someone 's at work , [ br ] they ca n't also be at a party . similarly , if we go back to the letter of recommendation case , [ br ] recall that a professor 's writing a letter of recommendation for a student who wants to apply for a prestigious award in philosophy , and the professor writes nothing but `` bob has wonderful penmanship . '' this communicates that the professor has nothing great to say about [ br ] bob 's ability in philosophy , because he violates the [ br ] first maxim of quantity , `` say as much as is required . '' clearly in a letter of recommendation for a prestigious philosophy award , much more is required [ br ] than information about the student 's penmanship . and again , he also infringes [ br ] on the maxim of relevance , because a student 's penmanship is not very relevant to [ br ] their philosophical ability . let 's look at one more example that we have n't touched on yet . suppose we had received [ br ] a package a few days ago and i put it away somewhere , and now you 're looking for it and you ask me `` where 's the package ? '' i ca n't remember where i put it , so i say `` i ca n't remember where [ br ] i put it , but i remember that it 's either in the [ br ] attic or the bedroom , '' and so i say `` it 's in [ br ] the attic or the bedroom . '' you 're gon na understand from that that i 'm not sure where the package is . why is that ? well , if i knew where the package is , by the first maxim of quantity , [ br ] i should have told you so . i could have said [ br ] something more informative . if i knew for sure it was in the attic , saying `` it 's in the [ br ] attic '' would have been the appropriate response . but in this case , giving [ br ] the most informative answer , obeying the first maxim of quantity , clashes with maxim of [ br ] quality which tells me `` do n't say things that i know to be false , '' or `` do n't say things for [ br ] which i lack evidence . '' since i do n't remember [ br ] where i put the package , i do n't have enough evidence to assert `` it 's in the attic , '' and so i tell you `` it 's in the attic or the bedroom , '' and you understand , again , [ br ] that i do n't know which one . on the other hand , imagine a situation in which we 're playing [ br ] a game of treasure hunt , and you 're looking for a [ br ] package that i 've hidden . you 're having a very , very hard [ br ] time succeeding at this game , and so i give you a hint to help you out . i say `` the package is in [ br ] the attic or the bedroom . '' these are the same words i [ br ] used in the last scenario , the same words that , in the last scenario , communicated that i did n't [ br ] know where the package was . but in this case , given that our conversational purpose is different ( we 're on a treasure [ br ] hunt and i have a reason for telling you something [ br ] less informative ) , it does n't communicate that i do n't know where the package is . so again , in this case , the [ br ] maxim of quantity tells me to tell you something less informative , because for our conversational purpose , the amount of information that 's required is something that 's less [ br ] than maximally informative , something that gives you a [ br ] hint at where the package is , but does n't tell you exactly where it is . so again , i 've used exactly the same words , but this time , by saying `` the package is in the [ br ] attic or the bedroom , '' i have n't communicated to you [ br ] that i do n't know where it is . so in all , we can see that [ br ] by appealing to the fact that conversations are [ br ] cooperative activites among rational agents , we can explain how so much gets [ br ] communicated , without thinking that our words themselves [ br ] actually mean different things in different contexts . subtitles by the amara.org community

and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studies the literal [ br ] meaning of the words or sentences that we use .

i am a little confused why discussing the nature of our language is relative to philosophy , and do the same maxims apply to the written language ?

( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studies the literal [ br ] meaning of the words or sentences that we use . but very often , we communicate more than the literal meanings [ br ] of the words we use . and this is one of the main things that pragmatics studies . for example , if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . but i did n't actually say that . i just stated two facts : one , `` i 'm going to montreal this week , '' and second , `` my mother lives in montreal . '' but everybody 's gon na naturally understand that those facts are connected . that in fact going to see my mother is my reason for going to montreal . here 's another example . suppose somebody asks me `` are you coming to the party on friday ? '' and i say `` i have to work . '' they 're gon na understand that [ br ] i ca n't make it to the party because i have to work . but again , i did n't say that . it could be that i have to [ br ] work earlier in the day . all i said was that i [ br ] have to work on friday . i did n't say anything about [ br ] it conflicting with the party . but you 're naturally gon na [ br ] understand my reply of `` i have to work on [ br ] friday '' as being a reason for me not coming to the party . in some cases , we can even [ br ] use the very same words in different situations and communicate completely [ br ] different things . for example , if a teacher [ br ] writes on a report card for a first grade student [ br ] '' bob has wonderful penmanship , '' it 's gon na communicate just that : bob is doing very well [ br ] at handwriting in class . that 's something you wan na [ br ] master in the first grade . on the other hand , suppose [ br ] a professor 's writing a letter of recommendation for one of her philosophy students applying for a prestigious award in philosophy . and all she writes in the letter is [ br ] '' bob has wonderful penmanship . '' well , in addition to [ br ] communicating something about bob 's penmanship , [ br ] that 's gon na communicate that the professor does n't think bob is a very good philosopher [ br ] or deserving of the award . herbert paul grice , a philosopher who lived from 1913 to 1988 , was the first who tried [ br ] to explain this phenomenon in his paper `` logic and conversation . '' and much of what he [ br ] said laid the foundation for the study of pragmatics today . grice invented the term `` implicature . '' an implicature is whatever is meant , but not literally said . things that are suggested , [ br ] implied , or hinted at . one very important kind of implicature that he talked about is [ br ] conversational implicature : implicatures that come about due to general features of conversation . and remember here again , [ br ] when we 're talking about conversations , we 're often [ br ] including things like books , letter-writing and so on . what grice observed is that , in general , conversations are cooperative efforts . people aim to understand [ br ] each other and be understood . they wan na give and receive information . they wan na influence each [ br ] other and be influenced . people in conversation [ br ] generally do n't say just a bunch of disconnected remarks . and even in the most [ br ] casual of conversations , there 's generally some sort of goal or purpose to the conversation . we do n't just idly say random things for no reason at all . grice took all these [ br ] observations and proposed that what we 're doing is sort [ br ] of tacitly following these rational rules of conversation . rational rules are [ br ] rules that people follow because we 're rational creatures , as opposed to , say , conventional rules . so for example , some countries drive on the left side of the [ br ] road and some countries , like ours , drive on the right . that 's just a conventional rule . one is not better than the other . but the fact that we [ br ] follow such rules at all , that we do n't just drive [ br ] in any direction we want , that 's a rational rule . that 's our way of [ br ] cooperating with each other . that 's how we get to where we 're going all in one piece , as opposed [ br ] to crashing into each other . grice summarized these observations , this idea that conversations [ br ] are cooperative activities among rational agents , [ br ] with his central rule that he called the [ br ] '' cooperative principle . '' the cooperative principle is `` make your conversational contribution such as is required , at the [ br ] stage at which it occurs , by the accepted purpose or [ br ] direction of the talk exchange in which you are engaged . grice further explained [ br ] the cooperative principle by giving four rules , or maxims , that people generally [ br ] follow in conversation . recall that these rules , or maxims , are not supposed to be conventional rules that we happen to follow [ br ] in conversations , but rules that govern rational [ br ] cooperative activity in general . the first is the maxim of quantity : make your contribution as [ br ] informative as is required for the current purposes of the exchange , and do not make your contribution more informative than is required . for example , if we 're talking about going to see a movie tonight and we 're trying to [ br ] decide what movie to see , and you ask me `` okay , what 's playing ? `` , following the first part [ br ] of the maxim of quantity , i should make my contribution [ br ] as informative as is required . and this just seems perfectly rational . i 'm going to read you , [ br ] say , all the times of the movies at our [ br ] local theater tonight , as opposed to just reading you the time of one movie that 's playing . but i 'm also gon na follow [ br ] the second maxim of quantity , `` do not make your [ br ] contribution more informative `` than is required . '' i 'm not , for example , [ br ] gon na tell you the times of all the movies playing [ br ] across the entire country . even though in some sense it 's an answer to your question of `` what [ br ] movies are playing tonight ? '' it 's more informative than is required by the conversational purpose . in the same way , this is [ br ] the sort of rule we follow in lots of cooperative activities . so for example , if we 're [ br ] fixing a car together and you need four screws [ br ] to screw in the next piece , i 'm not gon na hand you two [ br ] screws , which is not enough , or six screws , which is too many . the next maxim is the maxim of quality , which is simply `` try to [ br ] make your contribution `` one that is true , `` do not say what you believe to be false , `` and do not say that for which [ br ] you lack adequate evidence . '' so , back to the movie example , if we 're talking about going [ br ] to see a movie tonight and we 're trying to decide what to see and you ask me what 's playing , i should n't tell you something [ br ] that i know is not playing , something i know to be false . or i should n't tell you [ br ] a certain movie 's playing at a certain time if i [ br ] have n't looked it up yet , if i do n't have adequate evidence . again , this is a rule that we follow in regular cooperative activity [ br ] outside of conversation . it 's just something that rational people , when they 're trying to cooperate [ br ] with each other , will do . for example , if we 're trying [ br ] to bake a cake together and it 's time to add the sugar , i should n't give you the salt , [ br ] pretending that it 's sugar . the next maxim is the maxim of relation , which is simply `` be relevant : `` say things that are relevant , `` do n't say things that are irrelevant . '' so again , if we 're talking about going to see a movie tonight , i [ br ] should n't start telling you about a good book that i read , unless of course i signal [ br ] a change in conversation . similarly , if we 're trying to bake a cake and it 's time to add the sugar , i should n't pass you a [ br ] book , or even an oven mitt , even though the oven mitt may be relevant at a later point in the cake-making . the final maxim has to do not with the content of what we [ br ] say , but with the way in which you say it , and that 's the maxim of manner . more specifically , grice says [ br ] '' we should be perspicuous : `` avoid obscurity of [ br ] expression , avoid ambiguity , `` be brief , and be orderly . '' so , for example , if we 're talking again about going to see a movie tonight and a particular movie 's [ br ] playing at the local theater , i should just say so , [ br ] namely something like `` movie x is playing at our local theater . '' and not something like `` our local theater will be displaying a [ br ] series of still images on a reel tonight . '' if i say something using an [ br ] odd expression like that , you 're gon na start thinking i 'm trying to communicate something else , something like `` i do n't [ br ] think a real movie 's playing at the theater tonight . '' similarly , if we 're [ br ] baking a cake together and it 's time to add the sugar , i should just directly hand you the sugar . i should n't , for example , [ br ] draw you a treasure map to the sugar . these maxims , together with [ br ] the cooperative principle , explain how people communicate more than the literal meanings of [ br ] their words in conversation . people assume tacitly [ br ] that the speaker 's obeying all the maxims she can and [ br ] the cooperative principle . these facts , plus facts about the purpose of the conversation , the [ br ] context of the conversation , and the general things [ br ] we know about the world , our world knowledge , allow us to calculate these implicatures , these [ br ] things that are hinted at , suggested , or otherwise communicated over and above the literal [ br ] meanings of our words . so let 's return to the [ br ] examples from the beginning and see how grice will [ br ] explain how those things get communicated . recall that i said that if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . and now we can understand why . you assume i 'm being a cooperative conversational participant , [ br ] and only saying things that , for example , are [ br ] relevant to each other . i 'm not giving you a bunch [ br ] of disconnected remarks that are facts about the world . well , the only way that the fact that my mother lives in montreal is relevant to my going to montreal is that i 'm going to [ br ] montreal to visit my mother . there 's a similar explanation [ br ] for the next example . so again , suppose somebody asks me `` are you coming to the party on friday ? '' and i answer `` i have to work , '' and you understand that [ br ] i 'm not going to the party because i have to work . well again , you 're gon na assume that i 'm observing the [ br ] cooperative principle and all the maxims . and again , i 'd be infringing [ br ] on the maxim `` be relevant '' if my statement `` i have to work '' was n't somehow an answer to the question `` are you going to the party on friday ? '' i would also be infringing [ br ] on the maxim of quantity , `` be as informative as required , '' since i would n't have provided [ br ] an answer to the question if `` i have to work '' was n't [ br ] an answer to the question `` are you going to the party on friday ? '' and finally , we can employ a little bit of our world knowledge [ br ] there because we know that if someone 's at work , [ br ] they ca n't also be at a party . similarly , if we go back to the letter of recommendation case , [ br ] recall that a professor 's writing a letter of recommendation for a student who wants to apply for a prestigious award in philosophy , and the professor writes nothing but `` bob has wonderful penmanship . '' this communicates that the professor has nothing great to say about [ br ] bob 's ability in philosophy , because he violates the [ br ] first maxim of quantity , `` say as much as is required . '' clearly in a letter of recommendation for a prestigious philosophy award , much more is required [ br ] than information about the student 's penmanship . and again , he also infringes [ br ] on the maxim of relevance , because a student 's penmanship is not very relevant to [ br ] their philosophical ability . let 's look at one more example that we have n't touched on yet . suppose we had received [ br ] a package a few days ago and i put it away somewhere , and now you 're looking for it and you ask me `` where 's the package ? '' i ca n't remember where i put it , so i say `` i ca n't remember where [ br ] i put it , but i remember that it 's either in the [ br ] attic or the bedroom , '' and so i say `` it 's in [ br ] the attic or the bedroom . '' you 're gon na understand from that that i 'm not sure where the package is . why is that ? well , if i knew where the package is , by the first maxim of quantity , [ br ] i should have told you so . i could have said [ br ] something more informative . if i knew for sure it was in the attic , saying `` it 's in the [ br ] attic '' would have been the appropriate response . but in this case , giving [ br ] the most informative answer , obeying the first maxim of quantity , clashes with maxim of [ br ] quality which tells me `` do n't say things that i know to be false , '' or `` do n't say things for [ br ] which i lack evidence . '' since i do n't remember [ br ] where i put the package , i do n't have enough evidence to assert `` it 's in the attic , '' and so i tell you `` it 's in the attic or the bedroom , '' and you understand , again , [ br ] that i do n't know which one . on the other hand , imagine a situation in which we 're playing [ br ] a game of treasure hunt , and you 're looking for a [ br ] package that i 've hidden . you 're having a very , very hard [ br ] time succeeding at this game , and so i give you a hint to help you out . i say `` the package is in [ br ] the attic or the bedroom . '' these are the same words i [ br ] used in the last scenario , the same words that , in the last scenario , communicated that i did n't [ br ] know where the package was . but in this case , given that our conversational purpose is different ( we 're on a treasure [ br ] hunt and i have a reason for telling you something [ br ] less informative ) , it does n't communicate that i do n't know where the package is . so again , in this case , the [ br ] maxim of quantity tells me to tell you something less informative , because for our conversational purpose , the amount of information that 's required is something that 's less [ br ] than maximally informative , something that gives you a [ br ] hint at where the package is , but does n't tell you exactly where it is . so again , i 've used exactly the same words , but this time , by saying `` the package is in the [ br ] attic or the bedroom , '' i have n't communicated to you [ br ] that i do n't know where it is . so in all , we can see that [ br ] by appealing to the fact that conversations are [ br ] cooperative activites among rational agents , we can explain how so much gets [ br ] communicated , without thinking that our words themselves [ br ] actually mean different things in different contexts . subtitles by the amara.org community

and remember here again , [ br ] when we 're talking about conversations , we 're often [ br ] including things like books , letter-writing and so on . what grice observed is that , in general , conversations are cooperative efforts . people aim to understand [ br ] each other and be understood . they wan na give and receive information .

if i understand correctly , this video is trying to explain some maxims in languages to improve conversations between people , right ?

( intro music ) my name is karen lewis , and i 'm an assistant [ br ] professor of philosophy at barnard college , columbia university . and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studies the literal [ br ] meaning of the words or sentences that we use . but very often , we communicate more than the literal meanings [ br ] of the words we use . and this is one of the main things that pragmatics studies . for example , if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . but i did n't actually say that . i just stated two facts : one , `` i 'm going to montreal this week , '' and second , `` my mother lives in montreal . '' but everybody 's gon na naturally understand that those facts are connected . that in fact going to see my mother is my reason for going to montreal . here 's another example . suppose somebody asks me `` are you coming to the party on friday ? '' and i say `` i have to work . '' they 're gon na understand that [ br ] i ca n't make it to the party because i have to work . but again , i did n't say that . it could be that i have to [ br ] work earlier in the day . all i said was that i [ br ] have to work on friday . i did n't say anything about [ br ] it conflicting with the party . but you 're naturally gon na [ br ] understand my reply of `` i have to work on [ br ] friday '' as being a reason for me not coming to the party . in some cases , we can even [ br ] use the very same words in different situations and communicate completely [ br ] different things . for example , if a teacher [ br ] writes on a report card for a first grade student [ br ] '' bob has wonderful penmanship , '' it 's gon na communicate just that : bob is doing very well [ br ] at handwriting in class . that 's something you wan na [ br ] master in the first grade . on the other hand , suppose [ br ] a professor 's writing a letter of recommendation for one of her philosophy students applying for a prestigious award in philosophy . and all she writes in the letter is [ br ] '' bob has wonderful penmanship . '' well , in addition to [ br ] communicating something about bob 's penmanship , [ br ] that 's gon na communicate that the professor does n't think bob is a very good philosopher [ br ] or deserving of the award . herbert paul grice , a philosopher who lived from 1913 to 1988 , was the first who tried [ br ] to explain this phenomenon in his paper `` logic and conversation . '' and much of what he [ br ] said laid the foundation for the study of pragmatics today . grice invented the term `` implicature . '' an implicature is whatever is meant , but not literally said . things that are suggested , [ br ] implied , or hinted at . one very important kind of implicature that he talked about is [ br ] conversational implicature : implicatures that come about due to general features of conversation . and remember here again , [ br ] when we 're talking about conversations , we 're often [ br ] including things like books , letter-writing and so on . what grice observed is that , in general , conversations are cooperative efforts . people aim to understand [ br ] each other and be understood . they wan na give and receive information . they wan na influence each [ br ] other and be influenced . people in conversation [ br ] generally do n't say just a bunch of disconnected remarks . and even in the most [ br ] casual of conversations , there 's generally some sort of goal or purpose to the conversation . we do n't just idly say random things for no reason at all . grice took all these [ br ] observations and proposed that what we 're doing is sort [ br ] of tacitly following these rational rules of conversation . rational rules are [ br ] rules that people follow because we 're rational creatures , as opposed to , say , conventional rules . so for example , some countries drive on the left side of the [ br ] road and some countries , like ours , drive on the right . that 's just a conventional rule . one is not better than the other . but the fact that we [ br ] follow such rules at all , that we do n't just drive [ br ] in any direction we want , that 's a rational rule . that 's our way of [ br ] cooperating with each other . that 's how we get to where we 're going all in one piece , as opposed [ br ] to crashing into each other . grice summarized these observations , this idea that conversations [ br ] are cooperative activities among rational agents , [ br ] with his central rule that he called the [ br ] '' cooperative principle . '' the cooperative principle is `` make your conversational contribution such as is required , at the [ br ] stage at which it occurs , by the accepted purpose or [ br ] direction of the talk exchange in which you are engaged . grice further explained [ br ] the cooperative principle by giving four rules , or maxims , that people generally [ br ] follow in conversation . recall that these rules , or maxims , are not supposed to be conventional rules that we happen to follow [ br ] in conversations , but rules that govern rational [ br ] cooperative activity in general . the first is the maxim of quantity : make your contribution as [ br ] informative as is required for the current purposes of the exchange , and do not make your contribution more informative than is required . for example , if we 're talking about going to see a movie tonight and we 're trying to [ br ] decide what movie to see , and you ask me `` okay , what 's playing ? `` , following the first part [ br ] of the maxim of quantity , i should make my contribution [ br ] as informative as is required . and this just seems perfectly rational . i 'm going to read you , [ br ] say , all the times of the movies at our [ br ] local theater tonight , as opposed to just reading you the time of one movie that 's playing . but i 'm also gon na follow [ br ] the second maxim of quantity , `` do not make your [ br ] contribution more informative `` than is required . '' i 'm not , for example , [ br ] gon na tell you the times of all the movies playing [ br ] across the entire country . even though in some sense it 's an answer to your question of `` what [ br ] movies are playing tonight ? '' it 's more informative than is required by the conversational purpose . in the same way , this is [ br ] the sort of rule we follow in lots of cooperative activities . so for example , if we 're [ br ] fixing a car together and you need four screws [ br ] to screw in the next piece , i 'm not gon na hand you two [ br ] screws , which is not enough , or six screws , which is too many . the next maxim is the maxim of quality , which is simply `` try to [ br ] make your contribution `` one that is true , `` do not say what you believe to be false , `` and do not say that for which [ br ] you lack adequate evidence . '' so , back to the movie example , if we 're talking about going [ br ] to see a movie tonight and we 're trying to decide what to see and you ask me what 's playing , i should n't tell you something [ br ] that i know is not playing , something i know to be false . or i should n't tell you [ br ] a certain movie 's playing at a certain time if i [ br ] have n't looked it up yet , if i do n't have adequate evidence . again , this is a rule that we follow in regular cooperative activity [ br ] outside of conversation . it 's just something that rational people , when they 're trying to cooperate [ br ] with each other , will do . for example , if we 're trying [ br ] to bake a cake together and it 's time to add the sugar , i should n't give you the salt , [ br ] pretending that it 's sugar . the next maxim is the maxim of relation , which is simply `` be relevant : `` say things that are relevant , `` do n't say things that are irrelevant . '' so again , if we 're talking about going to see a movie tonight , i [ br ] should n't start telling you about a good book that i read , unless of course i signal [ br ] a change in conversation . similarly , if we 're trying to bake a cake and it 's time to add the sugar , i should n't pass you a [ br ] book , or even an oven mitt , even though the oven mitt may be relevant at a later point in the cake-making . the final maxim has to do not with the content of what we [ br ] say , but with the way in which you say it , and that 's the maxim of manner . more specifically , grice says [ br ] '' we should be perspicuous : `` avoid obscurity of [ br ] expression , avoid ambiguity , `` be brief , and be orderly . '' so , for example , if we 're talking again about going to see a movie tonight and a particular movie 's [ br ] playing at the local theater , i should just say so , [ br ] namely something like `` movie x is playing at our local theater . '' and not something like `` our local theater will be displaying a [ br ] series of still images on a reel tonight . '' if i say something using an [ br ] odd expression like that , you 're gon na start thinking i 'm trying to communicate something else , something like `` i do n't [ br ] think a real movie 's playing at the theater tonight . '' similarly , if we 're [ br ] baking a cake together and it 's time to add the sugar , i should just directly hand you the sugar . i should n't , for example , [ br ] draw you a treasure map to the sugar . these maxims , together with [ br ] the cooperative principle , explain how people communicate more than the literal meanings of [ br ] their words in conversation . people assume tacitly [ br ] that the speaker 's obeying all the maxims she can and [ br ] the cooperative principle . these facts , plus facts about the purpose of the conversation , the [ br ] context of the conversation , and the general things [ br ] we know about the world , our world knowledge , allow us to calculate these implicatures , these [ br ] things that are hinted at , suggested , or otherwise communicated over and above the literal [ br ] meanings of our words . so let 's return to the [ br ] examples from the beginning and see how grice will [ br ] explain how those things get communicated . recall that i said that if i tell you `` i 'm going to montreal this week . `` my mother lives there , '' you 're gon na understand [ br ] that i 'm going to montreal in order to visit my mother . and now we can understand why . you assume i 'm being a cooperative conversational participant , [ br ] and only saying things that , for example , are [ br ] relevant to each other . i 'm not giving you a bunch [ br ] of disconnected remarks that are facts about the world . well , the only way that the fact that my mother lives in montreal is relevant to my going to montreal is that i 'm going to [ br ] montreal to visit my mother . there 's a similar explanation [ br ] for the next example . so again , suppose somebody asks me `` are you coming to the party on friday ? '' and i answer `` i have to work , '' and you understand that [ br ] i 'm not going to the party because i have to work . well again , you 're gon na assume that i 'm observing the [ br ] cooperative principle and all the maxims . and again , i 'd be infringing [ br ] on the maxim `` be relevant '' if my statement `` i have to work '' was n't somehow an answer to the question `` are you going to the party on friday ? '' i would also be infringing [ br ] on the maxim of quantity , `` be as informative as required , '' since i would n't have provided [ br ] an answer to the question if `` i have to work '' was n't [ br ] an answer to the question `` are you going to the party on friday ? '' and finally , we can employ a little bit of our world knowledge [ br ] there because we know that if someone 's at work , [ br ] they ca n't also be at a party . similarly , if we go back to the letter of recommendation case , [ br ] recall that a professor 's writing a letter of recommendation for a student who wants to apply for a prestigious award in philosophy , and the professor writes nothing but `` bob has wonderful penmanship . '' this communicates that the professor has nothing great to say about [ br ] bob 's ability in philosophy , because he violates the [ br ] first maxim of quantity , `` say as much as is required . '' clearly in a letter of recommendation for a prestigious philosophy award , much more is required [ br ] than information about the student 's penmanship . and again , he also infringes [ br ] on the maxim of relevance , because a student 's penmanship is not very relevant to [ br ] their philosophical ability . let 's look at one more example that we have n't touched on yet . suppose we had received [ br ] a package a few days ago and i put it away somewhere , and now you 're looking for it and you ask me `` where 's the package ? '' i ca n't remember where i put it , so i say `` i ca n't remember where [ br ] i put it , but i remember that it 's either in the [ br ] attic or the bedroom , '' and so i say `` it 's in [ br ] the attic or the bedroom . '' you 're gon na understand from that that i 'm not sure where the package is . why is that ? well , if i knew where the package is , by the first maxim of quantity , [ br ] i should have told you so . i could have said [ br ] something more informative . if i knew for sure it was in the attic , saying `` it 's in the [ br ] attic '' would have been the appropriate response . but in this case , giving [ br ] the most informative answer , obeying the first maxim of quantity , clashes with maxim of [ br ] quality which tells me `` do n't say things that i know to be false , '' or `` do n't say things for [ br ] which i lack evidence . '' since i do n't remember [ br ] where i put the package , i do n't have enough evidence to assert `` it 's in the attic , '' and so i tell you `` it 's in the attic or the bedroom , '' and you understand , again , [ br ] that i do n't know which one . on the other hand , imagine a situation in which we 're playing [ br ] a game of treasure hunt , and you 're looking for a [ br ] package that i 've hidden . you 're having a very , very hard [ br ] time succeeding at this game , and so i give you a hint to help you out . i say `` the package is in [ br ] the attic or the bedroom . '' these are the same words i [ br ] used in the last scenario , the same words that , in the last scenario , communicated that i did n't [ br ] know where the package was . but in this case , given that our conversational purpose is different ( we 're on a treasure [ br ] hunt and i have a reason for telling you something [ br ] less informative ) , it does n't communicate that i do n't know where the package is . so again , in this case , the [ br ] maxim of quantity tells me to tell you something less informative , because for our conversational purpose , the amount of information that 's required is something that 's less [ br ] than maximally informative , something that gives you a [ br ] hint at where the package is , but does n't tell you exactly where it is . so again , i 've used exactly the same words , but this time , by saying `` the package is in the [ br ] attic or the bedroom , '' i have n't communicated to you [ br ] that i do n't know where it is . so in all , we can see that [ br ] by appealing to the fact that conversations are [ br ] cooperative activites among rational agents , we can explain how so much gets [ br ] communicated , without thinking that our words themselves [ br ] actually mean different things in different contexts . subtitles by the amara.org community

and today , i want to talk to [ br ] you about gricean pragmatics . pragmatics is the study of how people use language in real conversations , and in books and emails [ br ] and other sorts of media of language use . pragmatics , on the one hand , is distinguished from [ br ] semantics , on the other hand , which studies the literal [ br ] meaning of the words or sentences that we use .

do we need to use all maxims when talking , or is it up to the scenario ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ?

so is a sequence basically just a function where the input is limited to positive integers ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

what is an explicit sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what is the difference between finite and infinite sequence , as they both have similar functions ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens .

so , what is the difference between a function and a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens .

why is a sequence discrete and a function is continuous ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 .

how to calculate the coefficient of x^98 , x^99 , x^49 in the expasion of ( x-1 ) ( x-2 ) ( x-3 ) ( x-4 ) ... ... ... ( x-100 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what 's the difference between a set and a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots .

are recursive or explicit arithmetic sequences used more often ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

is it possible to have a non-infinite sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

also , does infinite mean that you do not have any limit of terms in your pattern and you can go on and on ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail .

are their any infinite number sequences where each number is irrational ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers .

what do the squigly parentheses mean ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

also , does that sideways 8 mean infinity ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function .

what is the difference between denoting a sequence and defining a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

are explicit and recursive formulas denotations or definitions ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

why is the second sequence k=1 instead of k=3 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 .

i thought the k= was to indicate where the sequence begins ( and , obviously there is no end , hence the infinity ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens .

can you use sigma notation to shorten an arithmetic sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out .

while k=3 , so 3-1=2 and 2+3=5 how come 7 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now .

i wish i knew : what 's the simplest possible ( or at least one very simple ) example of recursion ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to .

does it matter what the sub-letter is or is it something specific for different sequences ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail .

so all sequences have to be finite or infinite ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 .

so k is like a place holder for the value of the place number in the sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ?

a sequence is a function limited to a positive input , right ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 .

what is a sub 1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

so the infinite sign is ... right ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term .

and why is it necessary to define the first term of tge sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail .

is there any use of sequences over functions in real life ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing .

what does `` k one less than ... '' mean ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

where does the 1 + and the 3 come from in 1 + 3 ( k - 1 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

can n or k ever be negative ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

in the equation a sub k = a sub ( k-1 ) - ( -3 ) what happens when k = 0 or k = 1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

would it not be easier to write the second sequence in form a= -1=4k instead of a=3+4 ( k-1 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 .

what does the a sub k part mean ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

can a sequence go infinitely both ways ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what is the exact definition for sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 .

how would i write a recursive formula for something with a changing difference , like 6 7 9 12 16 where the difference goes up ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 .

why is n't an infinite notation noted { x , x+y , x+ ( y*2 ) } ... instead of { x , x+y , x+ ( y*2 ) ... } ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

i do n't think i 'm understanding the part where it says a^k=1+3 ( k-1 ) where did the 1 come from ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term .

for a recursive rule , in a sub k-1 , what number are you plugging in ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

could writing ( k-1 ) in the explicit function definitions be avoided if we just start counting when k=0 rather than k=1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what is the difference between a series and a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

could n't the sequence defined as `` finite '' be also infinite ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time .

how do you write an explicit general term for a recursive sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers .

so what if your difference was a negative integer ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers .

would you then write the formula : asubk=1-3 ( k-1 ) ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here .

0 sal defines the sequence recursively , but how does the recursive definition know to stop at 10 because the sequence stops at 10 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term .

why does sal call the sequence a sub 1 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 .

how do you find the nth term for a sequence 1,4,9,16 ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

is the definition of explicit and recursive is that explicit is used in a finite sequence and recursive is used in an infinite sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ?

how is a sequence a function limited to positive integers ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term .

what would this sequence be called : { -1 , -4 , -7 , -10 , ... } where it would be negative ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case .

what does the word recursive even mean ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 .

also , do the type of brackets matter when writing { a ( k ) } ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail .

are sequences essentially the same thing as sets ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

what is the difference between a sequence and a series ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

can a sequence not have a pattern ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

can an irrational be expressed as a sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 .

in other words , when does a sequence not go from the first term ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

for the first example , the rule could also be 3k-2 right ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 .

can someone please explain how to make an finite equation ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

that 's not an attractive color . let me write this in . this is an explicit function .

is it possible to write the same sequence with different `` equations '' ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 .

how do you find a good sequence ?

what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 plus 3 is 4 . 4 plus 3 is 7 . 7 plus 3 is 10 . and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . so we go to 3 , to 7 , to 11 , 15 . and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . but to show that this is infinite , to show that we keep this pattern going on and on and on , i 'll put three dots . this just means we 're going to keep going on and on and on . so we could call this an infinite sequence . now , there 's a bunch of different notations that seem fancy for denoting sequences . but this is all they refer to . but i want to make us comfortable with how we can denote sequences and also how we can define them . we could say that this right over here is the sequence a sub k for k is going from 1 to 4 , is equal to this right over here . so when we look at it this way , we can look at each of these as the terms in the sequence . and this right over here would be the first term . we would call that a sub 1 . this right over here would be the second term . we 'd call it a sub 2 . i think you get the picture -- a sub 3 . this right over here is a sub 4 . so this just says , all of the a sub k 's from k equals 1 , from our first term , all the way to the fourth term . now , i could also define it by not explicitly writing the sequence like this . i could essentially do it defining our sequence as explicitly using kind of a function notation or something close to function notation . so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . when k is 3 , we get 7 . so let 's see . when k is 3 , we added 3 twice . let me make it clear . so this was a plus 3 . this right over here was a plus 3 . this right over here is a plus 3 . so whatever k is , we started at 1 . and we added 3 one less than the k term times . so we could say that this is going to be equal to 1 plus k minus 1 times 3 , or maybe i should write 3 times k minus 1 -- same thing . and you can verify that this works . if k is equal to 1 , you 're going to get 1 minus 1 is 0 . and so a sub 1 is going to be 1 . if k is equal to 2 , you 're going to have 1 plus 3 , which is 4 . if k is equal to 3 , you get 3 times 2 plus 1 is 7 . so it works out . so this is one way to explicitly define our sequence with kind of this function notation . i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? well , i could say that this is equal to -- and people tend to use a . but i could use the notation b sub k or anything else . but i 'll do a again -- a sub k. and here , we 're going from our first term -- so this is a sub 1 , this is a sub 2 -- all the way to infinity . or we could define it -- if we wanted to define it explicitly as a function -- we could write this sequence as a sub k , where k starts at the first term and goes to infinity , with a sub k is equaling -- so we 're starting at 3 . and we are adding 4 one less time . for the second term , we added 4 once . for the third term , we add 4 twice . for the fourth term , we add 4 three times . so we 're adding 4 one less than the term that we 're at . so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . so this right over here is explicit . that 's not an attractive color . let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ? well , we can also define it , especially something like an arithmetic sequence , we can also define it recursively . and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms of the term before it . and so then we could write a sub k is equal to the previous term . so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . now , how does this make sense ? well , we 're defining what a sub 1 is . and if someone says , well , what happens when k equals 2 ? well , they 're saying , well , it 's going to be a sub 2 minus 1 . so it 's going to be a sub 1 plus 3 . well , we know a sub 1 is 1 . so it 's going to be 1 plus 3 , which is 4 . well , what about a sub 3 ? well , it 's going to be a sub 2 plus 3. a sub 2 , we just calculated as 4 . you add 3 . it 's going to be 7 . this is essentially what we mentally did when i first wrote out the sequence , when i said , hey , i 'm just going to start with 1 . and i 'm just going to add 3 for every successive term . so how would we do this one ? well , once again , we could write this as a sub k. starting at k , the first term , going to infinity with -- our first term , a sub 1 , is going to be 3 , now . and every successive term , a sub k , is going to be the previous term , a sub k minus 1 , plus 4 . and once again , you start at 3 . and then if you want the second term , it 's going to be the first term plus 4 . it 's going to be 3 plus 4 . you get to 7 . and you keep adding 4 . so both of these , this right over here is a recursive definition . we started with kind of a base case . and then every term is defined in terms of the term before it or in terms of the function itself , but the function for a different term .

let me write this in . this is an explicit function . and so you might say , well , what 's another way of defining these functions ?

is recursive method easier than the explicit method ?