# What is the 32nd term of the arithmetic sequence where a1 = 14 and a13 = -58?

Jul 17, 2015

The 32nd term of the sequence is $\textcolor{red}{- 172}$.

#### Explanation:

We know that ${a}_{1} = 14$ and ${a}_{13} = - 58$.

We can use the ${n}^{\text{th}}$ term rule for an arithmetic sequence:

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

${a}_{13} = {a}_{1} + \left(13 - 1\right) d$

$- 58 = 14 + 12 d$

$- 58 - 14 = 12 d$

$- 72 = 12 d$

$d = - \frac{72}{12}$

$d = - 6$

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

${a}_{32} = 14 + \left(32 - 1\right) \left(- 6\right) = 14 + 31 \left(- 6\right) = 14 - 186$

${a}_{32} = - 172$