7.9: Sums of Finite Geometric Series
Anna is on a progressive workout plan, every day she adds 5% to her exercise time. If she starts by exercising 15mins 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.
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 Khan Academy: Geometric series sum to figure out mortgage payments
Guidance
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 TI83/4 calculator:
We can use this formula as long as the series in question is geometric.
Example A
Find the sum of the first 10 terms of a geometric series with a_{1} = 3 and r = 5.
Solution
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 B
Find the sum of each series:
 a) The first term of a geometric series is 4, and the common ratio is 3. Find S_{8}.
 b) The first term of a geometric series is 80, and the common ratio is (1/4). Find S_{7}.
Solution
 a)
 b)
Example C
Prove the formula by induction
Solution
 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.
 2. Assume that the sum of the first k terms in a geometric series is .
 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.
Concept question wrapup "Anna is on a progressive workout plan, every day she adds 5% to her exercise time. If she starts by exercising 15mins on the first day, how many minutes will she have exercised all together on day 45?" Use the formula:


Vocabulary
A finite series has a defined ending value.
An infinte series does not have a defined ending value.
A geometric series is a series where the difference between terms increases or decreases between each pair of terms.
Guided Practice
Questions
1) Find the sum: (Hint: if a_{n} = 640 , what is n?)
2) Use a geometric series to answer the question:
 In January, a company’s sales totaled $11,000. It is predicted that the company’s sales will increase 5% each month for the next year. At this rate, what will be the total sales for the year?
3) Write the first 5 terms of the sequence:
4) Write the 3rd, 4th, and 6th terms of:
5) Find the sum of the series:
Solutions
1)
2)
3) Just do the multiplication for each term
 ..... for
 ..... for
 ..... for
 ..... for
 ..... for
 the first 5 terms are:
4) As with problem 3, just perform the operations on the indicated values of n:
 ..... for
 ..... for
 ..... for
 the 3rd, 4th, and 6th terms are:
5) To find the sum of the series

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

Or we can use the formula:
Practice
Find the sum of the finite series. You may simply calculate the individual terms and add them, or you may use the formula:
Image Attributions
Description
Learning Objectives
Here you will learn how to calculate the sum of a series when the difference between the members is not constant.