# How do you find the formula for a_n for the arithmetic sequence a_1=-4, a_5=16?

May 8, 2017

Set up and solve a system of equations using the definition of the arithmetic sequence. Answer: ${a}_{n} = - 4 + 5 \left(n - 1\right)$

#### Explanation:

We know that from the general formula for arithmetic sequences:
${a}_{n} = {a}_{1} + \left(n - 1\right) d$
where ${a}_{n}$ is the nth term, ${a}_{1}$ is the 1st term, $n$ is the term number, and $d$ is the difference between each term.

So, we write out what we know:
${a}_{1} = - 4$
${a}_{5} = 16$
Using the general formula for arithmetic sequences, the second equation becomes:
${a}_{5} = {a}_{1} + \left(5 - 1\right) d = 16$
By substituting ${a}_{1} = - 4$ into the above equation, we get:
$- 4 + 4 d = 16$
Now, we simply solve for d:
$- 1 + d = 4$
$d = 5$

Therefore, the formula for the given arithmetic sequence is:
${a}_{n} = - 4 + 5 \left(n - 1\right)$