Question

The sum of 0.2+0.004+0.00006+0.0000008+... to $\infty$ is

• A. $\frac{200}{891}$
• B. $\frac{2000}{9801}$
• C. $\frac{1000}{9801}$
• D. none of these

Answer 1

The correct option is B $\frac{2000}{9801}$

$S_n=\frac{2}{10}+\frac{4}{{10}^3}+\frac{6}{{10}^5}+...\infty$

$\Rightarrow \frac{S_n}{2}=\frac{1}{10}+\frac{2}{{10}^3}+\frac{3}{{10}^5}+...$

$\frac{S_n}{200}=\frac{1}{{10}^3}+\frac{2}{{10}^5}+\frac{3}{{10}^7}+...$

$\Rightarrow \frac{S_n}{2}-\frac{S_n}{200}=\frac{1}{10}+\frac{1}{{10}^3}+\frac{1}{{10}^5}+\frac{1}{{10}^7}+...$

$\Rightarrow \frac{99S_n}{200}=\frac{\frac{1}{10}}{1-\frac{1}{100}}$

$\Rightarrow \frac{99S_n}{200}=\frac{10}{99}$

$\therefore S_n=\frac{2000}{9801}$