Answer
The answer is $1140$.
Work Step by Step
For the first series ${a_n}=-5,10,-20,40,...$.
Common ratio $r=-2$.
This is the geometric series.
Sum of the first $9$ terms.
$S_{9}=\frac{-5(1-(-2)^{9})}{1-(-2)}$.
Simplify.
$S_{9}=-855$.
The second series is
$c_n=-2,1,-\frac{1}{2},\frac{1}{4}...$
Common ratio $r=-\frac{1}{2}$
This is the geometric series.
Sum of the infinite terms is
$S_{\infty}=\frac{-2}{1-\left ( -\frac{1}{2} \right)}$
$S_{\infty}=\frac{-2}{1+\frac{1}{2} }$
$S_{\infty}=\frac{-2}{\frac{2+1}{2} }$
$S_{\infty}=-\frac{4}{3}$
The product is $=(-855)\cdot (-\frac{4}{3})$.
$=1140$.
