Question

The sum of $1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+....$ up to $n$ terms is

• A. $\frac{25}{16}-\frac{4n+5}{16\times 5^{n-1}}$
• B. $\frac{3}{4}-\frac{2n+5}{16\times 5^{n+1}}$
• C. $\frac{3}{7}-\frac{3n+5}{16\times 5^{n-1}}$
• D. $\frac{1}{2}-\frac{5n+1}{3\times 5^{n+2}}$

Answer 1

The correct option is A $\frac{25}{16}-\frac{4n+5}{16\times 5^{n-1}}$

Let $S=1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+...\left(1\right)$

Multiplying $S$ with $\frac{1}{5}$ we get

$\frac{1}{5}S=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}+...\left(2\right)$

Subtracting $\left(2\right)$ from $\left(1\right)$

$S=1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+....T_n\underset{–}{-\frac{1}{5}S=-[\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+...T_{n-1}]-T_n}(1-\frac{1}{5})S=1+\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+.....-T_{n-1}(1-\frac{1}{5})S=\frac{\left(1\right)(1-\frac{1}{5^n})}{(1-\frac{1}{5})}-\frac{n}{5^{n-1}}\frac{4}{5}S=\frac{5}{4}-\frac{(4n+5)}{4\times 5^n}S=\frac{25}{16}-\frac{(4n+5)}{16\times 5^{n-1}}$