Question

Evaluate: $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....$ to infinite terms

• A. $\frac{25}{8}$
• B. $\frac{26}{8}$
• C. $\frac{27}{8}$
• D. None of the above

Answer 1

The correct option is D None of the above

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ .....(1)

Multiplying with $\frac{1}{5}$ on both sides,

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

$\therefore \left(1\right)-\left(2\right)$

$\Rightarrow \frac{4S}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+...$

$\Rightarrow \frac{4S}{5}=1+3[\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+...]$

$\Rightarrow \frac{4S}{5}=1+3\left[\frac{\frac{1}{5}}{1-\frac{1}{5}}\right]$

$\Rightarrow \frac{4S}{5}=1+3\left[\frac{1}{4}\right]$

$\therefore S=\frac{35}{16}$