# Converting recursive & explicit forms of arithmeticÂ sequences

CCSS Math: HSF.BF.A.2

Learn how to convert between recursive and explicit formulas of arithmetic sequences.

Before taking this lesson, make sure you know how to find recursive formulas and explicit formulas of arithmetic sequences.

# Converting from a recursive formula to an explicit formula

An arithmetic sequence has the following recursive formula.

Recall that this formula gives us the following two pieces of information:

- The first term is $\greenE 3$
- To get any term from its previous term,
**add $\maroonC 2$.**In other words, the common difference is $\maroonC 2$.

Let's find an explicit formula for the sequence.

Remember that we can represent a sequence whose first term is $\greenE A$ and common difference is $\maroonC B$ with the standard explicit form $\greenE A+\maroonC B(n-1)$.

Therefore, an explicit formula of the sequence is $a(n)=\greenE{3}\maroonC{+2}(n-1)$.

## Check your understanding

# Converting from an explicit formula to a recursive formula

### Example 1: Formula is given in standard form

We are given the following explicit formula of an arithmetic sequence.

This formula is given in the standard explicit form $\greenE A+ \maroonC B(n-1)$ where $\greenE A$ is the first term and that $\maroonC B$ is the common difference. Therefore,

- the first term of the sequence is $\greenE 5$, and
- the common difference is $\maroonC{16}$.

Let's find a recursive formula for the sequence. Recall that the recursive formula gives us two pieces of information:

- The first term $($which we know is $\greenE 5)$
- The pattern rule to get any term from the term that comes before it $($which we know is "add $\maroonC{16}$"$)$

Therefore, this is a recursive formula for the sequence.

### Example 2: Formula is given in simplified form

We are given the following explicit formula of an arithmetic sequence.

Note that this formula is

*given in the standard explicit form $\greenE A+ \maroonC B(n-1)$.***not**For this reason, we can't simply use the structure of the formula to find the first term and the common difference. Instead, we can find the first two terms:

Now we can see that the first term is $\greenE{12}$ and the common difference is $\maroonC{2}$.

Therefore, this is a recursive formula for the sequence.