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### Course: Operations and Algebraic Thinking 229+ > Unit 7

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

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**.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: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.

### 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.

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

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

*not*arithmetic because the difference between consecutive terms is not the same.### 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:

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

### 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?

- So if adding and subtracting from the previous terms create an arithmetic sequence, would multiplying or dividing make a geometric sequence?(29 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.(50 votes)

- 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.(55 votes)

- Are arithmetic sequences always either addition or subtraction(17 votes)
- Yes that is what makes them arithmetic. Multiply and divide are geometric sequences.(28 votes)

- this is so easy but Sal somehow made it look like it's some kind of a more complicated thing(14 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(10 votes)
- I thought it would be easier here with the videos(2 votes)

- can somebody please explain why we cannot multiply or divide sequentially and still have the sequence be arithmetic?(5 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.(13 votes)

- do all arithmetic sequences have to have real numbers?(5 votes)
- A sequence can be of unreal numbers I think that arithmetic progression should of real numbers(11 votes)

- I put that 3, 9, 27, and 81 was a arithmetic sequence. Why is that wrong? I know why but I just feel like that should be Arithmetic.(4 votes)
- In this case that is a geometric sequence in which you are multiplying or dividing by a number. An arithmetic sequence would be where you are adding or subtracting by the same number. In this case 3,9,27... is not arithemtic.(10 votes)

- Why does multiplying or dividing count as an arithmetic sequence?(1 vote)
- They don't.

An**arithmetic sequence**uses addition/subtraction of a common value to create the next term in the sequence.

A**geometric sequences**uses multiplication/division of a common value to create the next term in the sequence.

Hope this helps.(16 votes)

- If I multiple the last number by a fixed number in a sequence,is it not that an arithmetic sequences(5 votes)
- Simply put if its multiplied or divided it'll be geometric whereas if its added or subtracted its an arithmetic secuence(2 votes)