# What is the 32nd term of the arithmetic sequence where a1 = -33 and a9 = -121?

Jul 4, 2015

${a}_{32} = - 374$

#### Explanation:

An arithmetic sequence is of the form:
${a}_{i + 1} = {a}_{i} + q$

Therefore, we can also say:
${a}_{i + 2} = {a}_{i + 1} + q = {a}_{i} + q + q = {a}_{i} + 2 q$

Thus, we can conclude:
${a}_{i + n} = {a}_{i} + n q$

Here, we have:
${a}_{1} = - 33$
${a}_{9} = - 121 \rightarrow {a}_{1 + 8} = - 33 + 8 q = - 121$

$\rightarrow 8 q = - 121 + 33 = - 88 \rightarrow q = \frac{- 88}{8} = - 11$

Therefore:
${a}_{32} = {a}_{1 + 31} = - 33 - 11 \cdot 31 = - 33 - 341 = - 374$