Question

$S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ Find the value of 16S. ___

Answer 1

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ (1) Above given series is an AGP with common ratio $\frac{1}{5}$
Multiply by common ratio to equation 1.
$\frac{S}{5}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+...$ (2)
Subtract equation 2 from equation 1
$S-\frac{S}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+...$
$\frac{4S}{5}=1+3\left(\frac{\frac{1}{5}}{1-\frac{1}{5}}\right)$
$=1+3\left(\frac{\frac{1}{5}}{\frac{4}{5}}\right)=1+\frac{3}{4}=\frac{7}{4}$
$\frac{4S}{5}=\frac{7}{4}$
$S=\frac{35}{16}$,
16S=35