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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, you will learn an easy to use formula for summing an arithmetic sequence. Watch as we use the formula in action and then get a chance to use it yourself as you follow along.

Our video lesson begins with a quick overview of what an **arithmetic sequence** is. It is a string of numbers where the difference between each pair of successive numbers is the same. So, for example, our number line is an arithmetic sequence where the difference is 1 between each pair of the numbers. This difference between the pairs of numbers is referred to as the common difference. You can also think of an arithmetic sequence as telling you how much you have of something as it grows. Say you have a savings account at the bank. If you start with 10 dollars and you continue to put 10 dollars into the account every day, your account balance can be written as an arithmetic sequence where the difference between each pair of numbers is 10. We write our arithmetic sequence between curly brackets and with commas between the numbers. Writing our sequence for the bank, we have {10, 20, 30, 40, . . .}. We use the three dots at the end to show that the sequence goes on indefinitely.

Sometimes, we want to add up the numbers in our sequence. For example, if our arithmetic sequence shows how many strawberries we are able to harvest each day, then adding up our sequence will tell us how many total strawberries we will have picked. To do this, we have a formula that will help us. The formula says that the sum of the first *n* terms of our arithmetic sequence is equal to *n* divided by 2 times the sum of twice the beginning term, *a*, and the product of *d*, the common difference, and *n* minus 1. The *n* stands for the number of terms we are adding together.

The part of this formula that we will use is the part after the equals sign. The part before the equals sign just tells you that you are going to add each term. What this first part does is gives you a way to calculate each term in your sequence and then to sum it up. But since we are looking for what this sum equals, we only need to concern ourselves with the part after the equal sign.

To use this formula, we label our *n*, *a*, and *d* in our problem and then we plug the appropriate numbers in and evaluate. Let's see how this is done. Say we are the strawberry farmers, and we picked 20 strawberries the first day. This is strawberry season, so every day more and more strawberries are ripening. Each day, we are able to pick 5 more strawberries than the previous day. We can write this as an arithmetic sequence where the common difference is 5. We write {20, 25, 30, . . .}.

To find out how many strawberries we are able to pick after 30 days, we need to use our sum formula. We label our *n* as 30, our *a* as 20 (since that is our beginning number), and *d* as 5 (since that is our common difference). Now that everything is labeled, we can plug in these numbers and evaluate to get our answer. We plug our numbers into the part of the formula that is behind the equals sign. We get 30 divided by 2 times the sum of 2 times 20 and 30 minus 1 times 5.

We begin evaluating this expression by first subtracting the 30 minus 1 to get 29. We then multiply this 29 with the 5 to get 145. We then multiply the 2 with the 20 to get 40. We add the 40 to the 145 to get 185. Now we can divide the 30 by the 2 to get 15. Multiplying the 15 with the 185, we get 2775. So, after 30 days, we will have picked 2775 strawberries. That's a lot of strawberries! Just think of how many strawberry shortcakes you can make with that!

Now it's your turn. See if you can find the sum of this sequence as you follow along. Our sequence is {5, 7, 9, 11, . . .}. First, we need to make sure that this is an arithmetic sequence. We check to see what the difference is between our numbers. Is this difference the same between each pair of numbers? Yes, it is. We have an arithmetic sequence with a common difference of 2 and a beginning number of 5.

We want to find the sum of the first 10 terms of this sequence. How do we do this? Yes, we will use the sum formula. We will label our *n*, *a*, and *d* so we can plug these numbers into the formula to evaluate. We have our *n* as 10, our *a* as 5, and our *d* as 2. Plugging these in, we get 10 divided by 2 times 2 times 5 plus 10 minus 1 times 2. Evaluating, we get 10 divided by 2 times 2 times 5 plus 9 times 2. This becomes 10 divided by 2 times 2 times 5 plus 18. Now we have 10 divided by 2 times 10 plus 18. Next, we have 10 divided by 2 times 28. 10 divided by 2 becomes 5, so we have 5 times 28. This becomes our answer of 140.

We are done! Let's review now. An **arithmetic sequence** is a string of numbers where the difference between each pair of successive numbers is the same. This difference is also referred to as the common difference. The formula to find the sum of the first n terms of our sequence is *n* divided by 2 times the sum of twice the beginning term, *a*, and the product of *d*, the common difference, and *n* minus 1. The *n* stands for the number of terms we are adding together. To use this formula, we label our *n*, *a*, and *d* so we can plug them in and evaluate to get our answer.

Once you've completed this lesson, you'll be able to:

- Define arithmetic sequence and common difference
- Identify the arithmetic sum formula
- Explain how to calculate an arithmetic sum using this formula

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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
- 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
- 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
- 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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