# Ch 21: Mathematical Sequences and Series

### About This Chapter

## Mathematical Sequences and Series - Chapter Summary and Learning Objectives

Watch the videos in this chapter to learn how mathematical patterns can be used to solve problems that would be pretty difficult to calculate otherwise. Instructors introduce you to factorial and summation notation, as well as the formulas required to find particular terms in a sequence or determine the sum of a mathematical series. Once you've worked your way through this chapter, you should be able to:

- Find any term in an arithmetic or geometric sequence
- Identify the common difference in an arithmetic sequence
- Recognize the common ratio in a geometric sequence
- Add all terms in an arithmetic sequence
- Find the sum of an infinite geometric series
- Understand how special sequences are generated

Video | Objectives |
---|---|

What Is a Mathematical Sequence? | Learn how to recognize terms in a sequence. Explore the differences between arithmetic and geometric sequences. |

Introduction to Sequences: Finite and Infinite | Differentiate between a finite and infinite sequence. Get tips for finding a particular term in a sequence. |

How to Use Factorial Notation: Process and Examples | Learn what factorial notation is and explain how to use it. |

Summation Notation and Mathematical Series | Understand how to build a series formula used to solve arithmetic sequences. |

How to Use Series and Summation Notation: Process and Examples | Get instruction on the proper use of summation notation. |

Understanding Arithmetic Series in Algebra | Define an arithmetic series. Learn how to find the common difference in an arithmetic sequence. |

How to Calculate an Arithmetic Series | Discover formulas used to add all the terms in an arithmetic series. |

Arithmetic Sequences: Definition & Finding the Common Difference | Get tips for writing terms in an arithmetic sequence and finding the common difference. |

How and Why to Use the General Term of an Arithmetic Sequence | Apply the formula used to find the general term of an arithmetic sequence. |

The Sum of the First n Terms of an Arithmetic Sequence | Learn how to find the sum of the first n terms of an arithmetic sequence. |

How to Find and Classify an Arithmetic Sequence | Memorize the two formulas used to find any term in an arithmetic sequence. |

Working with Geometric Sequences | Understand how to write terms of a geometric sequence and find the common ratio. |

Finding and Classifying Geometric Sequences | Practice creating a formula to find terms of a geometric sequence. |

How and Why to Use the General Term of a Geometric Sequence | Explain how to use the formula for the general term of a geometric sequence. |

The Sum of the First n Terms of a Geometric Sequence | Apply the formula used to find the sum of the first n terms in a geometric sequence. |

Understand the Formula for Infinite Geometric Series | Learn how to find the sum of an infinite geometric series using the proper formula. |

How to Calculate a Geometric Series | Differentiate between the formulas used to calculate infinite and finite geometric series. |

Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences | Explain the uses of recursion formulas for arithmetic, algebraic and geometric sequences. |

Arithmetic and Geometric Series: Practice Problems | Complete practice problems requiring you to use the correct formula for arithmetic and geometric series. |

Using Sigma Notation for the Sum of a Series | Use sigma notation to signify the sum of a series. |

Mathematical Induction: Uses & Proofs | Learn what mathematical induction is. Explore its uses to prove statements. |

How to Find the Value of an Annuity | Understand how to find the value of an annuity. |

How to Use the Binomial Theorem to Expand a Binomial | Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion. |

Special Sequences and How They Are Generated | Find out how triangular numbers, tetrahedral numbers, cube numbers, square numbers and Fibonacci numbers are generated. |

### 1. What is a Mathematical Sequence?

Math is often the most fun when it acts like a puzzle to be solved. The branch of math where this is the most true involves sequences. Get an introduction to the basics and important vocabulary, as well as learn where sequences appear in nature!

### 2. Introduction to Sequences: Finite and Infinite

In this video lesson, we will learn about the many patterns that are possible in sequences. See how some sequences stop after a while and how some sequences never stop.

### 3. How to Use Series and Summation Notation: Process and Examples

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.

### 4. Understanding Arithmetic Series in Algebra

In this video lesson, we will learn about arithmetic series and how they work. Learn also how to find the pattern of an arithmetic sequence along with finding the sum.

### 5. How to Calculate an Arithmetic Series

In this lesson, you'll learn how to sum up any arithmetic sequence without having to add up each and every term. You'll never guess the circumstances that led to the discovery of this marvelous formula.

### 6. Arithmetic Sequences: Definition & Finding the Common Difference

Watch this video lesson to learn the pattern that defines an arithmetic sequence. Learn how to identify arithmetic sequences as well as find the pattern easily. You will also learn the formula you can use to write any number in your sequence.

### 7. How and Why to Use the General Term of an Arithmetic Sequence

Watch this video lesson to learn the formula for the general term of an arithmetic sequence. Also learn why this formula makes working with an arithmetic sequence much easier.

### 8. How to Find and Classify an Arithmetic Sequence

Arithmetic sequences are everywhere, and with a few tricks you learn here, you could end up looking like a psychic the next time you go to a movie or a football game!

### 9. Working with Geometric Sequences

In this video lesson, we'll learn how to recognize when a sequence of numbers is a geometric sequence, how to find the common ratio and how to expand a sequence to as many numbers as we want!

### 10. Finding and Classifying Geometric Sequences

Want your YouTube video to get a lot of hits? Besides including a cute baby or an adorable cat, getting your video to have a big common ratio is the key. Learn what I'm talking about here!

### 11. The Sum of the First n Terms of a Geometric Sequence

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.

### 12. Understand the Formula for Infinite Geometric Series

Watch this video lesson to learn how to calculate the sum of an infinite geometric series. Learn about the interesting thing that happens when your common ratio is less than one.

### 13. How to Calculate a Geometric Series

If you're not satisfied with knowing how many views your YouTube video will get any one day and need to know how many total views you're getting, geometric series are for you! Learn more in this lesson.

### 14. Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences

When dealing with sequences in math, both algebraic and geometric, we come across recursive rules. Watch this video lesson to learn how recursion works and how you can use a recursive rule to get to your next number using a previous number.

### 15. Arithmetic and Geometric Series: Practice Problems

There are many different types of problems concerning series and sequences - including some that are pretty abstract. From bouncing balls to adding up odd numbers, test your skills here with additional arithmetic and geometric series practice!

### 16. Using Sigma Notation for the Sum of a Series

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.

### 17. Mathematical Induction: Uses & Proofs

Watch this video lesson to learn about mathematical induction and how you can use it to prove mathematical statements. See how it is similar to falling dominoes.

### 18. How to Find the Value of an Annuity

Want to see how much money you will have in the future if you make a set payment every month towards your annuity? Then watch this video lesson for the formula and how to use it.

### 19. How to Use the Binomial Theorem to Expand a Binomial

In this video lesson, you will see what the binomial theorem has in common with Pascal's triangle. Learn how you can use Pascal's triangle to help you to easily expand a binomial.

### 20. Special Sequences and How They Are Generated

Sequences are interesting things in math and in nature. The most interesting thing is that these sequences have a pattern to how they are generated. We will learn about a few of them in this video lesson.

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### Other Chapters

Other chapters within the Precalculus: High School course

- Working With Inequalities
- Absolute Value Equations
- Working with Complex Numbers
- Introduction to Quadratics
- Working with Quadratic Functions
- Basics of Polynomial Functions
- Working with Higher-Degree Polynomials
- Graphing Piecewise Functions
- Understanding Function Operations
- Graph Symmetry
- Graphing with Functions
- Rate of Change
- Rational Functions & Difference Quotients
- Rational Expressions and Function Graphs
- Exponential Functions & Logarithmic Functions
- Using Trigonometric Functions
- Trigonometric Graphs
- Trigonometric Applications
- Solving Trigonometric Identities
- Vectors, Matrices and Determinants
- Sets in Algebra
- Analytic Geometry and Conic Sections
- Polar Coordinates and Parameterizations
- Continuity
- Limits
- Teacher Resources for Precalculus in High School