Answer
The equation for the $n$th term is
$a_n=5(-2)^{n-1}$
and $a_9=1280$
Work Step by Step
The given series is
$5,-10,20,-40,...$
First term $a_1=5$.
Common ratio $r=\frac{-10}{5}=-2$.
Equation for a geometric sequence is
$\Rightarrow a_n=a_1r^{n-1}$
Substitute $5$ for $a_1$ and $-2$ for $r$.
$\Rightarrow a_n=5(-2)^{n-1}$
$\Rightarrow a_n=5(-2)^{n-1}$
Substitute $9$ for $n$.
$\Rightarrow a_9=5(-2)^{9-1}$
Simplify.
$\Rightarrow a_9=5(-2)^{8}$
$\Rightarrow a_9=1280$.
