## Precalculus (6th Edition) Blitzer

$1,-3,9,-27,81,...$ Geometric.
When we observe the sequence carefully, we see that the difference between two consecutive terms is not the same. Thus, the common ratio will be computed as $1,-3,9,-27,81,...$ Now, ${{a}_{1}}=1,{{a}_{2}}=-3,{{a}_{3}}=9,{{a}_{4}}=-27,{{a}_{5}}=81,...$ So, \begin{align} & \frac{{{a}_{2}}}{{{a}_{1}}}=\frac{\left( -3 \right)}{1} \\ & =-3 \\ & \frac{{{a}_{3}}}{{{a}_{2}}}=\frac{9}{\left( -3 \right)} \\ & =-3 \end{align} \begin{align} & \frac{{{a}_{4}}}{{{a}_{3}}}=\frac{\left( -27 \right)}{9} \\ & =-3 \\ & \frac{{{a}_{5}}}{{{a}_{4}}}=\frac{81}{\left( -27 \right)} \\ & =-3 \end{align} Therefore, the common ratio is -3. Hence, the sequence is Geometric.