# Working with Geometric Sequences

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• 0:01 What is a Geometric Sequence?
• 1:24 Finding the Common Ratio
• 2:31 Continuing a Geometric…
• 3:11 More Examples
• 4:40 Lesson Summary

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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, we'll learn how to recognize when a sequence of numbers is a geometric sequence, how to find the common ratio and how to expand a sequence to as many numbers as we want!

## What Is a Geometric Sequence?

The defining pattern of a geometric sequence is actually quite easy to spot. This is because a geometric sequence is a sequence of numbers where each number is found by multiplying the previous number by a constant. For example, if our constant is 3, and the first number in our sequence is 1, then:

• The second number will be 1 * 3 = 3
• The third number will be 3 * 3 = 9
• The fourth number will be 9 * 3 = 27
• The fifth number will be 27 * 3 = 81
• The sixth number will be 81 * 3 = 243

We could keep going, but we'll stop there for now. We end up with this string of numbers: 1, 3, 9, 27, 81, 243.... Do you see how each number is 3 times the previous number?

You can think of a geometric sequence as a series of gift giving. For example, you start the ball rolling by giving a gift to three people. Then those three people keep the ball rolling by each giving a gift to three more people. Now there are 9 people receiving gifts. Then those 9 people each give a gift to three more people so that a total of 27 people are now receiving gifts. We could keep on going if we wanted to, but you probably get the idea.

## Finding the Common Ratio

Many times in math, you will be given a sequence of numbers and asked to determine if it is a geometric sequence. To do this, we simply divide each number in the sequence by the previous number. If the quotient of each division is the same, then we have a geometric sequence. This repeated quotient is called the common ratio, which is equal to the constant multiplied by each number to find the next number in the sequence.

Let's look at an example.

Say we have this sequence: 2, 4, 8, 16.... How can we determine if this a geometric sequence? We begin by dividing 4 by 2. What do we get? We get 2. Next, we divide 8 by 4. What do we get? We get 2 again. So far, so good. We are getting the same answer of 2. Next, we continue by dividing 16 by 8. What do we get here? We get 2 again. Great! So this is a geometric sequence with a common ratio of 2. This means that each number in the sequence is simply the previous number multiplied by 2.

## Continuing a Geometric Sequence

What if we wanted to expand a geometric sequence out to more numbers? Using our definition from earlier, we simply need to multiply our last number by the common ratio to find the next number in the sequence. We can repeat this process over and over to extend our sequence out as far as we'd like.

For example, if we wanted to continue our sequence of 2, 4, 8, and 16 to find the next number, we simply multiply the 16 by 2, which gives us 32. So 32 is my new last number. To find the next number, we multiply our 32 by 2 again. This next number is then 32 * 2, which is 64.

## More Examples

Let's practice with two more sequences. See if you can find the common ratio and continue the sequence as we work them out together.

3, 6, 12, 24...

First of all, is this a geometric sequence? When we divide each number in the sequence by the number preceding it, we find that we get the same result:

• 6 / 3 = 2
• 12 / 6 = 2
• 24 / 12 = 2

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