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Algebra II Textbook26 chapters | 256 lessons

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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!

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.

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.

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.

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

This means we have a geometric sequence with a common ratio of 2. We already have four terms in this sequence, but what if we wanted to find the sixth term? We would have to continue our sequence for two more numbers. To do this, we multiply the 24 by 2 to get 48, and then again, we multiply 48 by 2 to get 96. So 48 is our fifth term, and 96 is our sixth term.

Let's try another sequence.

1, 4, 16, 64

Dividing each number in this sequence by the number preceding it, we find that they all equal 4.

- 64 / 16 = 4
- 16 / 4 = 4
- 4 / 1 = 4

As we can see, this is a geometric sequence with a common ratio of 4. The fifth term of the sequence would be 64 * 4 = 256.

What have we learned? We've learned that a **geometric sequence** is a sequence of numbers where each number is found by multiplying the previous number by a constant. To determine if a sequence of numbers is a geometric sequence, we divide each number by the previous number. If each division has the same quotient, then it is a geometric sequence. The quotient is called the **common ratio**, which is equal to the constant multiplied by each number to find the next number in the sequence. To expand a geometric sequence, we multiply the last number of the sequence by the common ratio to find the next number. We can continue this process until we've reached our desired number.

Following this video lesson, you will be able to:

- Define geometric sequence and common ratio
- Explain how to find the common ratio and expand a geometric sequence

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Algebra II Textbook26 chapters | 256 lessons

- Introduction to Sequences: Finite and Infinite 4:57
- How to Use Factorial Notation: Process and Examples 4:40
- How to Use Series and Summation Notation: Process and Examples 4:16
- Arithmetic Sequences: Definition & Finding the Common Difference 5:55
- How and Why to Use the General Term of an Arithmetic Sequence 5:01
- The Sum of the First n Terms of an Arithmetic Sequence 6:00
- Understanding Arithmetic Series in Algebra 6:17
- Working with Geometric Sequences 5:26
- The Sum of the First n Terms of a Geometric Sequence 4:57
- Understand the Formula for Infinite Geometric Series 4:41
- Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences 5:52
- Using Sigma Notation for the Sum of a Series 4:44
- Mathematical Induction: Uses & Proofs 7:48
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