Anna is on a progressive workout plan, so every day she adds 5% to her exercise time. If she starts by exercising 15 mins on the first day, how many minutes will she have exercised all together on day 45?

This is a geometric series, since the difference between the exercise time on any two days is greater than the difference between any prior two days. You could just add: and so on up to 45, but that would be horribly tedious. In this lesson, you will learn how to answer a question like this will little effort.

### Sums of Finite Geometric Series

A finite geometric series is simply a geometric series with a specific number of terms. For example, consider the series: 50 + 25 + 12.5 + ....The series is geometric: the first term is 50, and the common ration is (1/2).

The sum of the first two terms is 50 + 25 = 75. We can write this as *S*_{2} = 75

The sum of the first three is 50 + 25 + 12.5 = 87.5. We can write this as *S*_{3} = 87.5

To find the value of *S*_{n} in general, we could simply add together the first *n* terms in a series. However, this would obviously be tedious for a large value of *n*. Given the regular pattern in a geometric series - every term is (1/*r*) of the previous term, and the *n*^{th} term is *a*_{n} = *a*_{1}*r*^{n - 1} , we can use induction to prove a formula for *S*_{n} .

The sum of the first *n* terms in a geometric series is

For example, for the series 50 + 25 + 12.5 + ... , the sum of the first 6 terms is:

The figure below shows the same calculation on a TI-83/4 calculator:

We can use this formula as long as the series in question is geometric.

### Examples

#### Example 1

Earlier, you were asked a question about Anna and her progressive workout plan.

Every day she adds 5% to her exercise time. If she starts by exercising 15 mins on the first day, how many minutes will she have exercised all together on day 45?

Use the formula:

minutes.

#### Example 2

Find the sum of the first 10 terms of a geometric series with *a*_{1} = 3 and *r* = 5.

The sum is 58,593.

Notice that because the common ratio in this series is 5, the terms get larger and larger. This means that for increasing values of *n* the sums will also get larger and larger. In contrast, in the series with common ratio (1/2), the terms gets smaller and smaller. This situation implies something important about the sum.

#### Example 3

Find the sum of each series:

- The first term of a geometric series is 4, and the common ratio is 3. Find
*S*_{8}.

- The first term of a geometric series is 80, and the common ratio is (1/4). Find
*S*_{7}.

#### Example 4

Prove the formula by induction.

Step 1) If *n* = 1, the *n*^{th} sum is the first sum, or *a*_{1} . Using the hypothesized equation, we get . This establishes the base case.

Step 2) Assume that the sum of the first *k* terms in a geometric series is .

Step 3) Show that the sum of the first *k*+1 terms in a geometric series is .

The sum is the sum, plus the term | |
---|---|

Substitute from step 2, and substitute the term | |

The common denominator is | |

Simplify the fraction | |

It is proven. |

Therefore we have shown that for a geometric series. Now we can use this equation to find any sum of a geometric series.

#### Example 5

Find the sum: (Hint: if *a*_{n} = 640 , what is *n*?).

#### Example 6

Write the first 5 terms of the sequence: .

Just do the multiplication for each term

..... for

..... for

..... for

..... for

..... for

the first 5 terms are:

#### Example 7

Write the 3rd, 4th, and 6th terms of: .

As with Example 6, just perform the operations on the indicated values of *n*:

..... for

..... for

..... for

the 3rd, 4th, and 6th terms are:

#### Example 8

Find the sum of the series: .

We could calculate all of the values for and add them, getting:

Or we can use the formula:

### Review

Find the sum of the finite series. You may simply calculate the individual terms and add them, or you may use the formula: .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.9.