Question

The sum of 1 + $\frac{2}{5}$ + $\frac{3}{\left(5^2\right)}$ + $\frac{4}{\left(5^3\right)}$ + .........upto n terms is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Given series, lets $S_n$ = 1 + $\frac{2}{5}$ + $\frac{3}{5^2}$ + $\frac{4}{5^3}$ + .........+$\frac{n}{5^{n-1}}$

$\frac{1}{5}$$S_n$ = $\frac{1}{5}$ + $\frac{2}{5^2}$ + $\frac{3}{5^3}$ + .........+$\frac{n}{5^n}$

Subtracting,

(1 - $\frac{1}{5}$)$S_n$ = 1 + $\frac{1}{5}$ + $\frac{1}{5^2}$ + $\frac{1}{5^3}$ + .........+upto n terms - $\frac{n}{5^n}$

⇒ $\frac{4}{5}$$S_n$ = $\frac{1-\frac{1}{5^n}}{\frac{4}{5}}$ - $\frac{n}{5^n}$ ⇒ $S_n$ = $\frac{25}{16}$ - $\frac{4n+5}{16\times 5^{n-1}}$