# The second term in an arithmetic sequence is 5 and the fifth term is 68. What is the function?

Jan 9, 2017

The general formula for a term can be written:

${a}_{n} = 21 n - 37$

#### Explanation:

The general term of an arithmetic sequence is given by the formula:

${a}_{n} = a + d \left(n - 1\right)$

where $a$ is the initial term and $d$ the common difference.

We find:

$3 d = \left(a + 4 d\right) - \left(a + d\right) = {a}_{5} - {a}_{2} = 68 - 5 = 63$

Dividing both ends by $3$ we get:

$d = 21$

Then:

${a}_{1} = a = \left(a + d\right) - d = {a}_{2} - d = 5 - 21 = - 16$

So the formula for the general term can be written:

${a}_{n} = - 16 + 21 \left(n - 1\right) = 21 n - 37$