# Geometric Series: Formula& Example

Instructor: Jennifer Beddoe
A geometric series is a series of numbers with a constant ratio between successive terms. Keep reading to discover more about geometric series, learn how to find the common ratio, and take a quiz to measure your understanding.

## Definitions

Let's define a few basic terms before jumping into the subject of this lesson. A series is a group of numbers. It can be a group that is in a particular order, or it can be just a random set. A geometric series is a group of numbers that is ordered with a specific pattern. The pattern is determined by multiplying a certain number to each number in the series. This determines the next number in the series. The number multiplied must be the same for each term in the series and is called the common ratio.

## Geometric Series

A geometric series is sometimes called a geometric sequence or a geometric progression. They all mean the same thing: a listing of numbers that follow a specific pattern. The pattern is regulated by the common ratio, which is the number that is the ratio between consecutive numbers in the series. If the series of numbers does not have a common ratio, then it is not a geometric series.

Geometric series have important practical applications in engineering, finance, biology, economics, computer science and other scientific fields. Some examples involving the use of geometric series include radioactive decay, decline in value of your car (or house) over time and determining the long-term results of actions taken today. It's like the old shampoo commercial, 'If you tell two friends, and they tell two friends, and they tell two friends....' - it's a geometric series.

The equation for a geometric series can be written as follows:

A, AR, AR^2, AR^3,....

A is the starting number, and R is the common ratio.

For example, write the geometric series of 4 numbers when A = 2 and R = 3

2 (the starting number), 6 (2 x 3), 18 (6 x 3 or 2 x 3^2), 54 (18 x 3 or 2 x 3^3)

So, the geometric series is:

2, 6, 27, 54

## Determining the Common Ratio

The common ratio is the amount between each number in a geometric series. It is called the common ratio because it is the same, or common, to each number, and it also is the ratio between two consecutive numbers in the series.

To determine the common ratio, you can just divide each number from the number preceding it in the series. For example, what is the common ratio in the following series of numbers?

{2, 4, 8, 16}

Starting with the number at the end of the series, divide by the number immediately preceding it.

16/8 = 2

Continue to divide to ensure that the pattern is the same for each number in the series.

8/4 = 2

4/2 = 2

Since the ratio is the same for each set, you can say that the common ratio is 2.

Therefore, you can say that the formula to find the common ratio of a geometric series is:

d = a(n) / a(n - 1)

Where a(n) is the last term in the sequence and a(n - 1) is the previous term in the series.

If you divide and find that the ratio between each number in the series is not the same, then there is no common ratio, and the series is not geometric.

## Examples

1.) What is the common ratio in the following series?

{3, 9, 27, 81}

81/27 = 3

27/9 = 3

9/3 = 3

The ratio between each of the numbers in the series is 3, therefore the common ratio is 3.

2.) What is the common ratio in the following series?

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