Question

The sum of $0.2+0.004+0.00006+0.0000008+\dots \infty$ , is

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

Answer 1

The correct option is D none of these

Let the sum be $S$,
$S=\frac{2}{10}+\frac{4}{{10}^3}+\frac{6}{{10}^5}+\frac{8}{{10}^7}\cdots \to \left(1\right)$
Dividing above equation by $100$,
$\frac{S}{100}=\frac{2}{{10}^3}+\frac{4}{{10}^5}+\frac{6}{{10}^7}+\cdots \to \left(2\right)$
Subtracting (2) from (1) by shifting one place, we get
$\frac{99S}{100}=\frac{2}{10}+\frac{2}{{10}^3}+\frac{2}{{10}^5}+\frac{2}{{10}^7}+\dots$
By using formula,
$a+ar+a{r}^2+\cdots =\frac{a}{1-r}$
Here $a=\frac{2}{10}$,r=$\frac{1}{100}$,
We get,
$\frac{99S}{100}=\frac{\frac{2}{10}}{1-\frac{1}{100}}$
$\therefore S=\frac{2000}{9801}$