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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.

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.

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.

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.

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.

Let's see how we can use this formula now. Let's say we are potato farmers. We want to know how many potatoes we have grown after harvesting for the fourth round. So our *n* is 4. We began with 1 potato, so our *a* is 1. Our common ratio is 20 because we multiplied by 20 each time to get our next number in our sequence. We have all the numbers we need to work our formula, so let's go ahead and use it now. We plug in all our values.

Now, we can go ahead and evaluate. We first take our 20 to the fourth power to get 160,000. 1 minus 160,000 is -159,999. 1 minus 20 is -19. -159,999 divided by -19 is 8,421. 8,421 times 1 is itself, 8,421. And there we have our answer.

Want to try one more? This time, you try to work it out as we work on it together. Let's find the sum of the first 6 terms of the geometric sequence that begins with the numbers 2, 4, and 8.

First, we need to find our common ratio. We do this by dividing each term by its previous term. Each time we divide, we should get the same number. We divide 4 by 2 and 8 by 4. What do we get? We get 2, so our common ratio, our *r*, is 2. We want to find the first 6 terms, so our *n* is 6. Our *a* is 2, since that is our beginning number. We can now fill in our formula with these values:

Now, we evaluate.

Our answer is 126, and we are done. How did you do?

Let's review what we've learned. A **geometric sequence** is a sequence where each term or number is the previous term or number multiplied by a certain constant. Our common ratio is our multiplication constant. The formula to find the sum of the first *n* terms of a geometric sequence is *a* times 1 minus *r* to the *nth* power over 1 minus *r* where *n* is the number of terms we want to find the sum for, *a* our beginning term of our sequence, and *r* our common ratio. To use it, we find our values, plug them in, and evaluate.

After reviewing this lesson you should be able to calculate the sum of a geometric sequence up to a designated point.

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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
- How and Why to Use the General Term of a Geometric Sequence 5:14
- The Sum of the First n Terms of a Geometric Sequence 4:57
- 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
- How to Find the Value of an Annuity 4:49
- How to Use the Binomial Theorem to Expand a Binomial 8:43
- Special Sequences and How They Are Generated 5:21
- Go to Algebra II: Sequences and Series

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