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Sums of Finite Geometric Series

Series with defined ending value has sum: Sn = (a1(1-r^n))/(1-r)

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Sums of Finite Geometric Series

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 S2 = 75

The sum of the first three is 50 + 25 + 12.5 = 87.5. We can write this as S3 = 87.5

To find the value of Sn 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 nth term is an = a1rn - 1 , we can use induction to prove a formula for Sn .

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 a1 = 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:

  1. The first term of a geometric series is 4, and the common ratio is 3. Find S8.

  1. The first term of a geometric series is 80, and the common ratio is (1/4). Find S7.

Example 4

Prove the formula by induction.

Step 1) If n = 1, the nth sum is the first sum, or a1 . 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 an = 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. 

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Vocabulary

TermDefinition
finite series A series is finite if it has a defined ending value.
geometric series A geometric series is a geometric sequence written as an uncalculated sum of terms.
induction Induction is a method of mathematical proof typically used to establish that a given statement is true for all positive integers.
infinite series An infinite series is the sum of the terms in a sequence that has an infinite number of terms.
series A series is the sum of the terms of a sequence.

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