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## Algebra 1

### Course: Algebra 1 > Unit 9

Lesson 1: Introduction to arithmetic sequences- Sequences intro
- Intro to arithmetic sequences
- Intro to arithmetic sequences
- Extending arithmetic sequences
- Extend arithmetic sequences
- Using arithmetic sequences formulas
- Intro to arithmetic sequence formulas
- Worked example: using recursive formula for arithmetic sequence
- Use arithmetic sequence formulas

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# Intro to arithmetic sequences

CCSS.Math:

Get comfortable with sequences in general, and learn what arithmetic sequences are.

Before you take this lesson, make sure you know how to add and subtract negative numbers.

## What is a sequence?

Here are a few lists of numbers:

- 3, 5, 7 ...
- 21, 16, 11, 6 ...
- 1, 2, 4, 8 ...

Ordered lists of numbers like these are called

**sequences**. Each number in a sequence is called a**term**.3, comma | 5, comma | 7, comma, point, point, point |
---|---|---|

\uparrow | \uparrow | \uparrow |

1, start superscript, start text, s, t, end text, end superscript, start text, space, t, e, r, m, end text | 2, start superscript, start text, n, d, end text, end superscript, start text, space, t, e, r, m, end text | 3, start superscript, start text, r, d, end text, end superscript, start text, space, t, e, r, m, end text |

Sequences usually have

**patterns**that allow us to predict what the next term might be.For example, in the sequence 3, 5, 7 ..., you always add

*two*to get the next term:start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | ||||
---|---|---|---|---|---|

3, comma | 5, comma | 7, comma, point, point, point |

The three dots that come at the end indicate that the sequence can be extended, even though we only see a few terms.

We can do so by using the pattern.

For example, the fourth term of the sequence should be nine, the fifth term should be 11, etc.

start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | ||||||
---|---|---|---|---|---|---|---|---|---|

3, comma | 5, comma | 7, comma | 9, comma | 11, comma, point, point, point |

### Check your understanding

**Extend the sequences according to their pattern.**

## What is an arithmetic sequence?

For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called

**arithmetic sequences**.In an arithmetic sequence, the difference between consecutive terms is always the same.

For example, the sequence 3, 5, 7, 9 ... is arithmetic because the difference between consecutive terms is always two.

start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|

3, comma | 5, comma | 7, comma | 9, comma, point, point, point |

The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.

start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|

21, comma | 16, comma | 11, comma | 6, comma, point, point, point |

The sequence 1, 2, 4, 8 ... is

*not*arithmetic because the difference between consecutive terms is not the same.start color #ed5fa6, plus, 1, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 4, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|

1, comma | 2, comma | 4, comma | 8, comma, point, point, point |

### Check your understanding

## The common difference

The

**common difference**of an arithmetic sequence is the constant difference between consecutive terms.For example, the common difference of 10, 21, 32, 43 ... is 11:

start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|

10, comma | 21, comma | 32, comma | 43, comma, point, point, point |

The common difference of –2, –5, –8, –11 ... is negative three:

start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6 | |||||
---|---|---|---|---|---|---|---|

minus, 2, comma | minus, 5, comma | minus, 8, comma | minus, 11, comma, point, point, point |

### Check your understanding

## What's next?

Learn about formulas of arithmetic sequences, which give us the information we need to find any term in the sequence.

## Want to join the conversation?

- is the lucas series series also an arithmetic sequence

eg. {1,1,2,3,5,8,13,21,34...} where asub(k)=asub(k-1)+asub(k-2)(19 votes)- NO. Take a look at the difference between the terms of the sequence. The difference between the terms is not constant (not the same), hence not an arithmetic sequence.(42 votes)

- So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?(23 votes)
- In short, yes.

Arithmetic is always adding or subtracting the same constant term or amount.

Geometric is always multiplying or dividing by the same constant amount.(36 votes)

- Are arithmetic sequences always either addition or subtraction(13 votes)
- Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.(18 votes)

- Instead of learning it in the book, my teacher says to learn on here but its hard when I'm a visual learner XD(9 votes)
- do all arithmetic sequences have to have real numbers?(6 votes)
- A sequence can be of unreal numbers I think that arithmetic progression should of real numbers(11 votes)

- If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences(6 votes)
- Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence(2 votes)

- why I want to go home?(6 votes)
- cuz school sucks(2 votes)

- Is there many kinds of different formulas to write explicit and Recursive equations or there just one?(4 votes)
- There is basically one formula, you just have to change the numbers.(5 votes)

- i understand it but i was wondering if there is an easier

way to solve arithemic sequences(5 votes)- you'll see more about shortcuts when you get used to the formulas, the next few lessons explain them.(1 vote)

- can somebody please explain why we cannot multiply or divide sequentially and still have the sequence be arithmetic?(2 votes)
- Some historical mathematician defined arithmetic sequences has being defined by addition/subtractions of a common value to get from one term to the next.

Geometric sequences were defined as using multiplication/division by a common value to get from one term to the next.(5 votes)