Finding The Common Ratio Of A Geometric Sequence

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Copyright Contributors: Brigette Banaszak, Jennifer Beddoe

Author

Author:

Brigette Banaszak

Brigette has a BS in Elementary Education and an MS in Gifted and Talented Education, both from the University of Wisconsin. She has taught math in both elementary and middle school, and is certified to teach grades K-8.

Instructor

Instructor:

Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Learn the definition of a common ratio in a geometric sequence and the common ratio formula. Also, see examples on how to find common ratios in a geometric sequence.

Table of Contents

• Geometric Sequence
• Common Ratio Examples
• Lesson Summary

Show FAQ

What is the common ratio formula?

The formula to find the common ratio of a geometric sequence is: r = n^th term / (n - 1)^th term

Divide each number in the sequence by its preceding number.

How do you calculate the common ratio?

To calculate the common ratio in a geometric sequence, divide the n^th term by the (n - 1)^th term. Start with the last term and divide by the preceding term. Continue to divide several times to be sure there is a common ratio.

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• 0:02 Definition of Common Ratio
• 0:31 Determining the Common Ratio
• 1:46 Examples
• 3:24 Lesson Summary

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Geometric Sequence

A sequence is a series of numbers, and one such type of sequence is a geometric sequence. So, what is a geometric sequence?

A geometric sequence is a series of numbers that increases or decreases by a consistent ratio. Each term in the geometric sequence is created by taking the product of the constant with its previous term.

The general form of a geometric sequence where first term a, and in which each term is being multiplied by the constant r to find the next consecutive term, is:

a, ar, ar2, ar3, ar4, ar5, . . .

Geometric sequence

What Is a Common Ratio?

A common ratio of a geometric sequence is a constant that multiplies or divides each term to form the next term. In the geometric sequence a, ar, ar2, ar3, ar4, ar5, . . ., the r is the common ratio. Each term is multiplied by the common ratio, r, to form the next term.

How to Find Common Ratio

Here is how one can find the common ratio in a geometric sequence: The common ratio is calculated by dividing any term in the geometric sequence, n, by the preceding term, n - 1. Its formula for the geometric sequence a, ar, ar2, ar3, ar4, ar5, . . ., is

{eq}r = \frac{n\ term}{n - 1\ term} {/eq}

The common ratio, r, is

{eq}r = \frac{ar}{a} = \frac{ar^2}{ar} = \frac{ar^3}{ar^2} = \frac{ar^4}{ar^3} . . . {/eq}

Common ratio formula

If dividing the nth term by the (n-1)th term does not give a constant, then the sequence is not a geometric sequence.

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Common Ratio Examples

Here are some examples of how to find the common ratio of a geometric sequence:

Example 1

What is the common ratio for the geometric sequence: 2, 6, 18, 54, 162, . . .

To find the common ratio for this geometric sequence, divide the nth term by the (n-1)th term. Start with the term at the end of the sequence and divide it by the preceding term.

{eq}162 \div 54 = 3 {/eq}

Continue dividing, in the same way, to be sure there is a common ratio.

{eq}54 \div 18 = 3 \\ 18 \div 6 = 3 \\ 6 \div 2 = 3 {/eq}

Since the ratio is the same each time, the common ratio for this geometric sequence is 3.

Example 2

Find the common ratio for the geometric sequence: 3840, 960, 240, 60, 15, . . .

To find the common ratio for this geometric sequence, divide the nth term by the (n-1)th term. Start off with the term at the end of the sequence and divide it by the preceding term.

{eq}15 \div 60 = 0.25 {/eq}

Continue dividing, in the same way, to ensure that there is a common ratio.

{eq}60 \div 240 = 0.25 \\ 240 \div 960 = 0.25 \\ 3840 \div 960 = 0.25 {/eq}

Since the ratio is the same each time, the common ratio for this geometric sequence is 0.25.

Example 3

What is the common ratio for the sequence: 10, 20, 30, 40, 50, . . .

To find the common ratio for this sequence, divide the nth term by the (n-1)th term. Start off with the term at the end of the sequence and divide it by the preceding term.

{eq}50 \div 40 = 1.25 {/eq}

Continue dividing, in the same way, to be sure there is a common ratio.

{eq}40 \div 30 = 1.3333... \\ 30 \div 20 = 1.5 \\ 20 \div 10 = 2 {/eq}

Here, there is no common ratio. Each ratio is different, so this is not a geometric sequence.

Example 4

Find the common ratio for the geometric sequence: 3, -15, 75, -375, 1875, . .

To find the common ratio for this geometric sequence, divide the nth term by the (n-1)th term. Start off with the term at the end of the sequence and divide it by the preceding term.

{eq}1875 \div -375 = -5 {/eq}

Continue dividing, in the same way, to get a common ratio.

{eq}-375 \div 75 = -5 \\ 75 \div -15 = -5 \\ -15 \div 3 = -5 {/eq}

Since the ratio is the same each time, the common ratio for this geometric sequence is -5.

Example 5

What is the common ratio for the geometric sequence: 4, 10, 25, 62.5, 156.25, . . .

To find the common ratio for this geometric sequence, divide the nth term by the (n-1)th term. Begin with the term at the end of the sequence and divide it by the preceding term.

{eq}156.25 \div 62.5 = 2.5 {/eq}

Continue dividing, in the same way, to be sure there is a common ratio.

{eq}62.5 \div 25 = 2.5 \\ 25 \div 10 = 2.5 \\ 10 \div 4 = 2.5 {/eq}

Since the ratio is the same each time, the common ratio for this geometric sequence is 2.5.

The common ratio for a geometric sequence can be a positive number or a negative number, a whole number or a decimal, a number greater than 1, or less than 1.

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Lesson Summary

Features of a geometric sequence are:

• series of numbers increases or decreases by a constant ratio.
• Each term is multiplied by the constant ratio to determine the next term in the sequence.

The constant ratio of a geometric sequence:

• found by dividing a term by the preceding term
• the formula to find the common ratio for the geometric sequence a, ar, ar2, ar3, ar4, ar5, . . ., is

{eq}r = \frac{n\ term}{n - 1\ term} {/eq}

The common ratio for a geometric sequence can be a positive number or a negative number, a whole number or a decimal, a number greater than 1, or less than 1.

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Video Transcript

Definition of Common Ratio

Let's define a few basic terms before jumping into the subject of this lesson. A sequence 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 sequence is a group of numbers that is ordered with a specific pattern. The pattern is determined by a certain number that is multiplied to each number in the sequence. This determines the next number in the sequence. The number multiplied must be the same for each term in the sequence and is called a common ratio.

Determining the Common Ratio

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

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

{2, 4, 8, 16}

Starting with the number at the end of the sequence, 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 sequence 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 sequence.

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

Examples

Let's take a look at a few examples.

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

{3, 9, 27, 81}

81 / 27 = 3

27 / 9 = 3

9 / 3 = 3

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

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

{5, 10, 15, 20}

20 / 15 = 1.3

15 / 10 = 1.5

10 / 5 = 2

There is no common ratio. Since all of the ratios are different, there can be no common ratio.

3.) What is the common ratio in the following sequence?

{17, 25.5, 38.25, 57.375}

57.375 / 38.25 = 1.5

38.25 / 25.5 = 1.5

25.5 / 17 = 1.5

The common ratio does not have to be a whole number; in this case, it is 1.5.

4.) What is the common ratio in the following sequence?

{4, -12, 36, -108}

The common ratio also does not have to be a positive number. In this series, the common ratio is -3. When you multiply -3 to each number in the series you get the next number.

Lesson Summary

A geometric sequence is a sequence of numbers that is ordered with a specific pattern. Each successive number is the product of the previous number and a constant. The constant is the same for every term in the sequence and is called the common ratio. You can also think of the common ratio as a certain number that is multiplied to each number in the sequence. You can determine the common ratio by dividing each number in the sequence from the number preceding it. If the same number is not multiplied to each number in the series, then there is no common ratio.

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