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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 math, we have a notation for a lot of things. When you need to sum up a series, we also have a notation for that. You will learn about this notation, called Sigma notation, in this video lesson.

In math, in addition to numbers by themselves, we also have series of numbers where you have a string of numbers. Some of these strings of numbers end after just a few numbers, while other strings never end. You can have strings of numbers with a pattern as well as random strings.

In this video lesson, we will talk about the strings of numbers that have a pattern. If there is a pattern, then we can write the pattern as a formula. Once we have a formula, we can use our special notation when we want to add up our string of numbers, our series.

For example, say we have 2, 4, 6, ... for our series. We see that we have a pattern. Our pattern is that we add two every time we go to the next number. Our third number is 2 + 2 + 2 or 2 + 2*2. Our fourth number is 2 + 2 + 2 + 2 or 2 + 2*3. We can write this pattern as a formula, such as:

This formula will give us the numbers in our sequence. For example, the fourth term will be *a* sub 4 = 2 + (4 - 1)2; *a* sub 4 stands for the fourth term. So, for our current series of 2, 4, 6, our fourth term will be 2 + (4 - 1) or 3 * 2. So, we have 2 + 6, which is 8. Our fourth term is 8.

Since we are dealing with math, we sometimes want to add up our terms to see what kind of totals we get. We are usually interested in the total of a limited number of terms. For example, we might want to know the total of just the first six terms or the first 20 terms. For our pattern of 2, 4, 6, etc, we can add up our first five terms by writing 2 + 4 + 6 + 8 + 10 and then performing these additions.

Of course, with this being math, we like to write things as concisely as possible. We like to substitute symbols for more complicated things to make it easier to work with. For our summation, we have our **Sigma notation**, which looks like a big sideways M.

This notation tells us to add up our series.

Let's look at how to use this notation.

We notice that our Sigma notation, our sideways M, has an *n* = 1 on the bottom and a 5 on the top. This is telling us to plug our *n* value into the formula to the right of the Sigma notation starting with *n* = 1, then moving onto *n* = 2, *n* = 3, all the way up to the top number, *n* = 5.

We plug in our *n* value for each *n*, and then, we add all of the resulting terms. For *n* = 1, we have 2 + (1 - 1)2 or 2. For *n* = 2, we have 2 + (2 - 1)2 or 4. For *n* = 3, we have 2 + (3 - 1)2 or 6. For *n* = 4, we have 2 + (4 - 1)2 or 8. And lastly for *n* = 5, we have 2 + (5 - 1)2 or 10.

Now we go ahead and add up all these terms. We have 2 + 4 + 6 + 8 + 10, which equals 30. We have found the total for the first five terms of our series.

Let's try another example.

The information on the bottom and top of the Sigma notation tell you to calculate the sum of the formula to the right of the notation for *n* = 1, *n* = 2, *n* = 3, and *n* = 4. Writing it all out and plugging it in, we have this.

See how we've plugged in *n* = 1, *n* = 2, *n* = 3, and *n* = 4 and then added them all up? Now we go ahead and evaluate what we have. We end up with 6 + 8 + 10 + 12, which equals 36. And we are done.

Now let's review what we've learned. We've learned that the notation for summing is called the **Sigma notation**, and it looks like a sideways M. This notation will have little numbers on the bottom and top of it telling you to plug in these values into the formula for the series to the right of the notation. You begin with the bottom number and go up to the top number. After plugging in each of these little values into your function, you add up all these terms to get to your final answer.

Study this video lesson, then test your capacity to:

- Calculate the next number in a sequence
- Explain the purpose of sigma notation
- Use sigma notation
- Interpret and solve an equation that uses sigma notation

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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
- 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
- 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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