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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.

Have you ever seen a sideways M on a math test and wondered what it meant? In this video lesson, learn about the notation that we use to show that we want to sum something up. Learn how to use it and see some examples of it in use.

**Sequences** are strings of numbers with a pattern. Examples of sequences include our counting numbers, our even numbers, our odd numbers and so on. Anything with a pattern is a sequence. We can even count by 10s and we would have a sequence because we have a pattern. We can also have a sequence where we multiply each number by the same number, 2, for example, to get to the next number. We write our sequences inside curly brackets with commas between the numbers. For example, we write our odd numbers as {1, 3, 5, 7, . . .} and our multiplication by 2s as {1, 2, 4, 8, . . .}.

If we take the sum of a sequence, we have what is called a **series**. We take the sum of a sequence when we want to find out the total of a sequence. This sum is useful in terms of statistics. When we take the sum, we are simply adding our numbers in our sequence together. So for our sequence of odd numbers, the series would be 1 + 3 + 5 + 7 and so on.

While we can write out our series using addition, it is easier to use the **Sigma notation** to represent our series. The Sigma notation is another way to say 'sum.' It is also referred to as summation notation. This notation looks like a big sideways M. On the bottom of the symbol, we have little letters that tell us our beginning point. On the top of the symbol, there is a little number that tells us our end point. To the right of the symbol, we have a formula for our sequence. For example, our counting numbers have the simple formula of *n*.

Let's go ahead and use this Sigma notation now. We see that our beginning point is *n* = 1. So, to find our first number that we will be adding to, we plug in this value of *n* into the formula to the right of the Sigma symbol. To find the second number to add, we increase our value of *n* by 1, so that now *n* = 2, and we plug this value into the formula. We repeat this process for the next number, increasing *n* by 1 again, so now *n* = 3. We keep repeating until we reach the number at the top of the symbol, when *n* = 10. That is the last number we are adding. We end up with the series 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. To finish, we add up all these numbers to get our answer of 55.

Let's do another problem.

Here we have the Sigma notation again. We begin with *n* = 1 and keep adding until we reach *n* = 5, plugging each value of *n* into our formula as we go. Our first term is 2*n* = 2(1) = 2. Our second term is 2(2) = 4, our third term is 2(3) = 6, our fourth term is 2(4) = 8 and our fifth term is 2(5) = 10. Hey, isn't this the sequence of even numbers? Now we add 2 + 4 + 6 + 8 + 10 to get 30 for our answer.

Let's review what we've learned now. **Sequences** are strings of numbers with a pattern. **Series** are the sum of sequences. The notation for sum is called the **Sigma notation**. It looks like a big sideways M with little letters and numbers on the top and bottom. The bottom gives you your start point and the top gives you your end point. Immediately to the right of the Sigma symbol is the formula for our sequence.

To use the Sigma notation, we plug our starting point into the formula and evaluate to find our first number. To find the second number to add, we increase our starting point value by 1, and then we plug that into the formula. To find the third and so forth, we keep increasing by 1 until we reach the number that is at the top of the Sigma symbol. Once we've found all the numbers to add, we add to find our answer.

Once you are done with this lesson you should be able to:

- Explain what sequences and series are
- Describe Sigma notation and what it is used for
- Calculate the series of a number set

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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
- 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
- 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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