The Sum of the First n Terms of a Geometric Sequence

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  • 0:01 A Geometric Sequence
  • 1:21 The Sum Formula
  • 2:18 Using the Sum Formula
  • 3:19 Another Example
  • 4:11 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 how you can find the total of a geometric sequence up to a certain point. Learn the formula that you can use to help you find your answer.

A Geometric Sequence

In this video lesson, we are going to talk about geometric sequences and how to find the total of a geometric sequence up to a certain point. We call this the sum of the first n terms of a geometric sequence. Recall that a geometric sequence is a sequence where each term or number is the previous term or number multiplied by a certain constant. Each sequence has its own constant of multiplication, or common ratio.

An example of a geometric sequence happening in real life could be when you're growing potatoes. Suppose every potato you plant will give you 20 potatoes when it's done growing. And if you plant each of those 20 potatoes, you will get 20 new potatoes from each of those potatoes. So you end up with 20 * 20, or 400 potatoes. And if you plant these 400 potatoes, you will end up with 400 * 20 more potatoes.

For this potato example, our geometric sequence is looking like 1, 20, 400, and so on. Each number is the previous number multiplied by a certain constant, in our case 20. If you are the farmer, you might want to know how many potatoes you have grown over time. You want to add up all the potatoes you have grown over time. This is when you would use our handy formula for finding the sum of the first n terms of a geometric sequence.

The Sum Formula

This formula begins with the sigma notation, which tells you that you will be adding up a series of numbers. Next comes the formula for finding each term of our geometric sequence. This part is simply a times r to the kth power. Our sigma notation tells us to add these terms, starting with k = 0 all the way up to n - 1.

All of this is equal to a times 1 minus r to the nth power over 1 minus r. This tells us that we can use the part after the equal sign to find our sum of our first n terms. We don't even have to add the individual terms. This is very helpful if we want to find the sum of a large number of terms.

geometric sequence

In this formula, a stands for our beginning term or number, r is our common ratio, and n is the number of terms we want to find the sum for.

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