Question

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

Answer 1

Answer: 305

$S=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\dots$
$\frac{1}{5}S=\frac{7}{5^2}+\frac{9}{5^3}+\frac{13}{5^4}+\dots$
On subtracting
$\frac{4}{5}S=\frac{7}{5}+\frac{2}{5^2}+\frac{4}{5^3}+\frac{6}{5^4}+\dots$
$S=\frac{7}{4}+\frac{1}{10}\left(1+\frac{2}{5}+\frac{3}{5^2}+\dots \right)$
$S=\frac{7}{4}+\frac{1}{10}{\left(1-\frac{1}{5}\right)}^{-2}$
$=\frac{7}{4}+\frac{1}{10}\times \frac{25}{16}=\frac{61}{32}$
$\Rightarrow 160S=5\times 61=305$