# Arithmetic sequences review

CCSS Math: HSF.BF.A.2, HSF.IF.A.3, HSF.LE.A.2

Review arithmetic sequences and solve various problems involving them.

## Parts and formulas of arithmetic sequences

In arithmetic sequences, the difference between consecutive terms is always the same. We call that difference

**the common difference**.For example, the common difference of the following sequence is $+2$:

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

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

Arithmetic sequence formulas give $a(n)$, the $n^{\text{th}}$ term of the sequence.

This is the

**explicit**formula for the arithmetic sequence whose first term is $\blueD k$ and common difference is $\maroonC d$:This is the

**recursive**formula of that sequence:*Want to learn more about arithmetic sequences? Check out this video.*

## Extending arithmetic sequences

Suppose we want to extend the sequence $3,8,13,...$ We can see each term is $\maroonC{+5}$ from the previous term:

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

$3,$ | $8,$ | $13,...$ |

So we simply add that difference to find that the next term is $18$:

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

$3,$ | $8,$ | $13,$ | $18,...$ |

*Want to try more problems like this? Check out this exercise.*

## Writing recursive formulas

Suppose we want to write a recursive formula for $3,8,13,...$ We already know the common difference is $\maroonC{+5}$. We can also see that the first term is $\blueD3$. Therefore, this is a recursive formula for the sequence:

*Want to try more problems like this? Check out this exercise.*

## Writing explicit formulas

Suppose we want to write an explicit formula for $3,8,13,...$ We already know the common difference is $\maroonC{+5}$ and the first term is $\blueD3$. Therefore, this is an explicit formula for the sequence:

*Want to try more problems like this? Check out this exercise.*