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How and Why to Use the General Term of a Geometric Sequence

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  • 0:01 A Geometric Sequence
  • 1:37 The General Term
  • 2:23 Why We Use It
  • 3:07 How to Use It
  • 4:20 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.

Watch this video lesson to learn the formula for the general term of a geometric sequence. You will learn how to write your own general term given a particular geometric sequence as well as the reason for doing this.

A Geometric Sequence

In this video lesson, we are going to be talking about geometric sequences and how to use the formula for the general term of a geometric sequence. We learn about this because we encounter geometric sequences in real life, and sometimes we need a formula to help us find a particular number in our sequence. We define our geometric sequence as a series of numbers, where each number is the previous number multiplied by a certain constant. This constant varies for each series but remains constant within one series.

Wondering how this kind of series relates to the real world? You might be surprised to learn that this kind of thing happens all the time, and you might even be involved in it yourself. Imagine that you wanted to spread the message that Friday is the new red scarf day, and everyone should wear a red scarf. First, you send your message to five of your friends. So, we started off with one person, and now we have five new people.

In the message, you ask your friends to send this message about the new red scarf day to five of their friends to help you spread the word. You also ask them to keep the ball rolling and ask their friends to each spread the word to five of their friends. Now we have 25 new people who know about the message. If each of these 25 people tells 5 of their friends, how many new people will know about the red scarf day? One hundred twenty-five people. So we have a geometric sequence of 1, 5, 25, 125... because each number is the previous multiplied by a constant, the common ratio. The common ratio in our case is five. Each of our numbers is multiplied by five to get to the next.

The General Term

We actually have a formula that we can use to help us calculate the general term, or nth term, of any geometric sequence. It is x sub n equals a times r to the n - 1 power.

geometric sequence general term

In this formula, x sub n stands for the particular number in that sequence. So x sub 4 stands for the fourth term in our sequence. The n stands for the term that we are looking at. If n is 10, then we are looking for the tenth term in our sequence. The r stands for the common ratio, the multiplication constant that is used in the geometric sequence to calculate each successive number or term. We plug in our a and r for our geometric sequence when we want to use this formula.

Why We Use It

Why do we use this formula when we can just write our sequence to find that particular term that we want? Well, this works well for terms that are in the beginning of our series. We can easily find our third, fourth, and fifth terms by just writing out our sequence. For our red scarf example, we already have the first four terms.

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