How To Know in a Application Sequence/Series Problem Which Variable is $a_{0}$ or $a_{1}$? How do you determine what value takes on $a_{0}$ or $a_{1}$ in order to use either the infinite sum formula or $a_{n}$ formula?

For example, consider the following two problems:


A certain culture initially contains 10,000 bacteria and increases by 20% every hour.


(a) Find a formula for the number N(t) of bacteria present after t hours.

In this case, $10,000$ would be taken on by $a_{0}$, so the formula would be $N(t)=10000*(1.2)^{10}$. However, when I initially solved the problem, I thought that $10,000$ was $a_{1}$ and got $N(t)=10000*(1.2)^{10-1}$. 

How can I distinguish between these in a word problem? Is $10,000=a_{0}$ because nothing has been "done" to it yet? But $a_{1}$ is the "starting value" in a sequence, so why isn't $10000=a_{1}$?


A certain drug has a half life of 2 hours in the bloodstream. The drug is formulated to be administered in doses of D milligrams every 4 hours, but D is yet to be determined.


Find a formula to get the milligrams of drug in the bloodstream after     the $n$th dose. Show that this sum is $\frac{4}{3}D$.

Going by the logic used in the last problem, $D$ would be the starting value, or $a_{0}$. However, that's not the case because the answer is $a_{n}=D*(1/4)^{n-1},$ so clearly $D$ is $a_{1}$, giving the sequence $D+\frac{1}{4}D+...+\frac{1}{4}D^{n-1}$, where $D=a_{1}$. Why is this the case? Also, the sum is $\frac{a_{1}}{1-r}=\frac{4}{3}D$, again with $D=a_{1}$. 

Why is $D$ in this case $a_{1}$ rather than $a_{0}$? I know this may seem rather obvious to some of you but I'm really confused and I need to know this for my test.

Thank you!
The difference between the first and second problems you have listed is the starting point.  Think about each problem logically ... the first problem is asking how much of something there is if it grows at a rate of 20% each hour.  If zero hours have passed, zero growth has happened, so you will start at zero.  

The second problem ... again, thinking logically you are starting with 4 and working back down towards zero.  You would get closer and closer to zero without ever reaching zero.  (Poorly written word problem because it is essentially saying that the drug NEVER leaves your system ... which is creepy to imply).

Also, you mention that "a of 1" is the starting value in a sequence and your statement is correct but incomplete.  Yes, the starting value is 1 unless otherwise specified.  Your problems "otherwise specify" the starting points through the context of the word problems.
The difference between the first and second problems you have listed is the starting point.  Think about each problem logically ... the first problem is asking how much of something there is if it grows at a rate of 20% each hour.  If zero hours have passed, zero growth has happened, so you will start at zero.  

The second problem ... again, thinking logically you are starting with 4 and working back down towards zero.  You would get closer and closer to zero without ever reaching zero.  (Poorly written word problem because it is essentially saying that the drug NEVER leaves your system ... which is creepy to imply).

Also, you mention that "a of 1" is the starting value in a sequence and your statement is correct but incomplete.  Yes, the starting value is 1 unless otherwise specified.  Your problems "otherwise specify" the starting points through the context of the word problems.