# What is the 32nd term of the arithmetic sequence where a_1 = 15 and a_13= –57?

Aug 13, 2016

${a}_{32} = - 171$

#### Explanation:

The general formula for a term of an arithmetic sequence is:

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

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

So:

$- 72 = - 57 - 15 = {a}_{13} - {a}_{1} = \left(a + 12 d\right) - \left(a + 0 d\right) = 12 d$

Hence:

$d = - 6$

$a = {a}_{1} = 15$

So:

${a}_{32} = a + 31 d = 15 - 6 \cdot 31 = - 171$