# Intro to arithmetic sequences

CCSS Math: HSF.IF.A.3

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**.$3,$ | $5,$ | $7,...$ |
---|---|---|

$\uparrow$ | $\uparrow$ | $\uparrow$ |

$\footnotesize 1^\text{st}\text{ term}$ | $\footnotesize 2^\text{nd}\text{ term}$ | $\footnotesize 3^\text{rd}\text{ term}$ |

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:$\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | |||
---|---|---|---|---|

$3,$ | $5,$ | $7,...$ |

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.

$\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | |||||
---|---|---|---|---|---|---|---|---|

$3,$ | $5,$ | $7,$ | $9,$ | $11,...$ |

## 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.

$\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$3,$ | $5,$ | $7,$ | $9,...$ |

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

$\footnotesize\maroonC{-5\,\Large\curvearrowright}$ | $\footnotesize\maroonC{-5\,\Large\curvearrowright}$ | $\footnotesize\maroonC{-5\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$21,$ | $16,$ | $11,$ | $6,...$ |

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

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

$\footnotesize\maroonC{+1\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+2\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+4\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$1,$ | $2,$ | $4,$ | $8,...$ |

## 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:

$\footnotesize\maroonC{+11\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+11\,\Large\curvearrowright}$ | $\footnotesize\maroonC{+11\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$10,$ | $21,$ | $32,$ | $43,...$ |

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

$\footnotesize\maroonC{-3\,\Large\curvearrowright}$ | $\footnotesize\maroonC{-3\,\Large\curvearrowright}$ | $\footnotesize\maroonC{-3\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$-2,$ | $-5,$ | $-8,$ | $-11,...$ |

## 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.