# Intro to arithmetic sequence formulas

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

Get comfortable with the basics of explicit and recursive formulas for arithmetic sequences.

Before taking this lesson, make sure you know the basics of arithmetic sequences and have some experience with evaluating functions and function domain.

# What is a formula?

We are used to describing arithmetic sequences like this:

But there are other ways. In this lesson, we'll be learning two new ways to represent arithmetic sequences:

**recursive formulas**and**explicit formulas**. Formulas give us instructions on how to find any term of a sequence.To remain general, formulas use $n$ to represent any term number and $a(n)$ to represent the $n^\text{th}$ term of the sequence. For example, here are the first few terms of the arithmetic sequence 3, 5, 7, ...

$n$ | $a(n)$ |
---|---|

(The term number) | (The $n^\text{th}$ term) |

$1$ | $3$ |

$2$ | $5$ |

$3$ | $7$ |

We mentioned above that formulas give us instructions on how to find any term of a sequence. Now we can rephrase this as follows:

*formulas tell us how to find $a(n)$ for any possible $n$.*## Check your understanding

# Recursive formulas of arithmetic sequences

Recursive formulas give us two pieces of information:

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

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

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

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

$a(1)$ | $=\blueD3$ | ||

$a(2)$ | $=a(1)+2$ | $=\blueD3+2$ | $=\purpleC5$ |

$a(3)$ | $=a(2)+2$ | $=\purpleC5+2$ | $=\greenD7$ |

$a(4)$ | $=a(3)+2$ | $=\greenD7+2$ | $=\goldD9$ |

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

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

## Check your understanding

Now it's your turn to find terms of sequences using their recursive formulas.

Just as we used $a(n)$ to represent the $n^\text{th}$ term of the sequence 3, 5, 7, ..., we can use other letters to represent other sequences. For example, we can use $b(n)$, $c(n)$, or $d(n)$.

# Explicit formulas of arithmetic sequences

Here is an explicit formula of 3, 5, 7, ...

This formula allows us to simply plug in the number of the term we are interested in to get the value of that term.

In order to find the fifth term, for example, we need to plug $n=5$ into the explicit formula.

Lo and behold, we get the same result as before!

## Check your understanding

# Sequences are functions

Notice that the formulas we used in this lesson work like

*functions*: We input a term number $n$, and the formula outputs the value of that term $a(n)$.Sequences are in fact defined as functions. However, $n$ cannot be any real number value. There's no such thing as the negative fifth term or the 0.4th term of a sequence.

This means that the domain of sequences—which is the set of all possible inputs of the function—is the

*positive integers*.### A note about notation

We've been writing $a(4)$, for example, to represent the fourth term, but other sources sometimes write $a_4$.

Both notations are fine to use. We prefer $a(4)$ because it emphasizes that sequences are functions.