# Sequences, Probability, and Counting Theory

## Series and Their Notations


### Learning Objectives

1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Objective 1: Use summation notation to write a sum. (IA 12.1.5)

A series is the sum of the terms of a sequence. For example,  1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,  
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:

For the series, 1 + 6 + 11 + 16  + 21 + 26 + 31 + .... the summation notation is  

### Practice Makes Perfect


Use summation notation to write the sum.


### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)


### Practice Makes Perfect

A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

### Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The  of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation 
 represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, 
  to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series 
  for 
  through 
  We can begin by substituting the terms for 
 and listing out the terms of this series.
We can find the sum of the series by adding the terms:

### Using the Formula for Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference, 
 The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first 
 terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first 
  terms of an arithmetic series, we can derive a formula for the sum of the first 
  terms of any arithmetic series.
Because there are 
 terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first 
  terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.

### Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, 
  We can write the sum of the first 
  terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first 
  terms of a geometric series. We will begin by multiplying both sides of the equation by 
  
Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for 
 divide both sides by 
 
Use the formula to find the indicated partial sum of each geometric series.

### Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first 
 terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is 
 
This series can also be written in summation notation as 
 where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

### Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined.  The terms in this series approach 0:
The common ratio 
 
 As 
  gets very large, the values of 
 get very small and approach 0.   Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with 
 approach 0; the sum of a geometric series is defined when 
 
Determine whether the sum of the infinite series is defined.

### Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first 
 terms of a geometric series.
We will examine an infinite series with 
 What happens to 
  as 
  increases?
The value of 
  decreases rapidly. What happens for greater values of 
 
As 
  gets very large, 
 gets very small. We say that, as 
  increases without bound, 
 approaches 0. As 
  approaches 0, 
 approaches 1. When this happens, the numerator approaches 
  This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.

### Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with 
 and 
 After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after 
 deposits using the formula for the sum of the first 
 terms of a geometric series. In 6 years, there are 72 months, so 
 We can substitute 
 into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of 
  This means that because of the annuity, the parent earned $720.44 interest in their college fund.

### Key Equations


### Key Concepts

1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first 
 terms of an arithmetic series can be found using a formula. See  and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first 
 terms of a geometric series can be found using a formula. See  and .
7. The sum of an infinite series exists if the series is geometric with 
 
8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first 
 terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first 
  terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

### Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series 
  

### Numeric

For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first  terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first 
  terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

### Extensions


### Real-World Applications
