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High School Algebra I: Help and Review25 chapters | 292 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

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.

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**.

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.

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.

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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High School Algebra I: Help and Review25 chapters | 292 lessons

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