Question

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots \infty .$ Then the value of $S$ is

• A. $\frac{31}{16}$
• B. $\frac{17}{8}$
• C. $\frac{7}{4}$
• D. $\frac{35}{16}$

Answer 1

The correct option is D $\frac{35}{16}$

$S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\dots \frac{S}{5}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+\dots$

Subtracting the above two equations,
$\frac{4S}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+\dots \Rightarrow \frac{4S}{5}=1+\frac{3}{5}\left(\frac{1}{1-\frac{1}{5}}\right)\therefore S=\frac{35}{16}$