Question

If $S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\cdots ,$ then $160S$ is equal to

Answer 1

$S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\cdots \infty .\dots \left(1\right)$
$\Rightarrow \frac{1}{5}S=\frac{7}{5^2}+\frac{9}{5^3}+\frac{13}{5^4}+\cdots \infty .\dots \left(2\right)$
By $\left(1\right)-\left(2\right)$
$\Rightarrow \frac{4S}{5}=\frac{7}{5}+\frac{2}{5^2}+\frac{4}{5^3}+\frac{6}{5^4}+\cdots$
$\Rightarrow \frac{4S}{5}=\frac{7}{5}+M\cdots \left(3\right)$
Where, $M=\frac{2}{5^2}+\frac{4}{5^3}+\frac{6}{5^4}+\cdots \infty .\dots \left(4\right)$
$\Rightarrow \frac{1}{5}M=\frac{2}{5^3}+\frac{4}{5^4}+\frac{6}{5^5}+\cdots \infty .\dots \left(5\right)$
By $\left(4\right)-\left(5\right)$
$\frac{4M}{5}=\frac{2}{5^2}+2(\frac{1}{5^3}+\frac{1}{5^4}+\cdots )$
$\Rightarrow \frac{4M}{5}=\frac{2}{5^2}+\frac{1/5^3}{1-1/5}$
$\Rightarrow \frac{4M}{5}=\frac{1}{10}$
$\Rightarrow M=\frac{1}{8}$
Now, from equation $\left(1\right)$, $S=\frac{61}{32}$
$\Rightarrow 160S=305$