Answer
Product of the sum of first 6 terms of $a_{n}$ and the sum of all terms of infinite geometric series $c_{n}$ = -620
Work Step by Step
Given sequence
|$a_{n}$| = -5, 10, -20, 40......... This is a geometric sequence with common ratio = -2
By $a_{n}$ = a $r^{n-1}$.
$a_{6}$ = (-5) $(-2)^{6-1}$ = (-5) $(-2)^{5}$ = (-5)$\times$(-32) = 160
The sum of first 6 terms of $a_{n}$ =$\frac{6}{2}$($a_{1}$ + $a_{6}$)
= $\frac{6}{2}$(- 5 + 160 )
= 3$\times$ (155) = 465
|$c_{n}$| = -2, 1, $-\frac{1}{2}$, $\frac{1}{4}$.............. This is a geometric sequence with common ratio = $-\frac{1}{2}$
The sum of the all terms of infinite geometric series $c_{n}$ = $\frac{c_{1}}{1 - r}$
= $\frac{-2}{1 - ( -\frac{1}{2})}$
= $\frac{-2}{1 + \frac{1}{2}}$
= $\frac{-2}{\frac{3}{2}}$
= $-\frac{4}{3}$
Product of the sum of first 6 terms of $a_{n}$ and the sum of all terms of $c_{n}$ = 465$\times$($-\frac{4}{3}$) = -620
