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Common Ratio: Definition & Concept

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The common ratio is the ratio between two numbers in a geometric sequence. Keep reading for a detailed definition, the formula for determining the common ratio and some example problems. A quiz at the end will allow you to test your knowledge.
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

 
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