# Recursive formulas for arithmetic sequences

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

Learn how to find recursive formulas for arithmetic sequences. For example, find the recursive formula of 3, 5, 7,...

Before taking this lesson, make sure you are familiar with the basics of arithmetic sequence formulas.

# How recursive formulas work

Recursive formulas give us two pieces of information:

- The first term of the sequence
- The pattern rule to get any term from the term that comes before it

Here is a recursive formula of the sequence $3, 5, 7,...$ along with the interpretation for each part.

In the formula, $n$ is any term number and $a(n)$ is the $n^\text{th}$ term. This means $a(1)$ is the first term, and $a(n-1)$ is the term before the $n^\text{th}$ term.

In order to find the fifth term, for example, we need to extend the sequence term by term:

$a(n)$ | $=a(n\!-\!\!1)+2$ | ||
---|---|---|---|

$a(1)$ | $=\greenE 3$ | ||

$a(2)$ | $=a(1)+2$ | $=\greenE 3+2$ | $=\purpleC5$ |

$a(3)$ | $=a(2)+2$ | $=\purpleC5+2$ | $=\blueD 7$ |

$a(4)$ | $=a(3)+2$ | $=\blueD 7+2$ | $=\goldD9$ |

$a(5)$ | $=a(4)+2$ | $=\goldD9+2$ | $=11$ |

Cool! This formula gives us the same sequence as described by $3,5,7,...$

## Check your understanding

# Writing recursive formulas

Suppose we wanted to write the recursive formula of the arithmetic sequence $5, 8, 11,...$

The two parts of the formula should give the following information:

- The first term $($which is $\greenE 5)$
- The rule to get any term from its previous term $($which is "add $\maroonC{3}$"$)$

Therefore, the recursive formula should look as follows: