Question

The sum of the series $S=1+2\left(\frac{10}{11}\right)+3{\left(\frac{10}{11}\right)}^2+\cdots \text{upto}\infty$ is equal to

• A. $121$
• B. $120$
• C. $111$
• D. $110$

Answer 1

The correct option is A $121$

Let $\frac{10}{11}=x.$
Then $S=1+2x+3x^2+\cdots \text{upto}\infty \cdots \left(1\right)$
$xS=x+2x^2+\cdots \text{upto}\infty \cdots \left(2\right)$
Subtracting $\left(1\right)$ and $\left(2\right)$,
$S(1-x)=1+x+x^2+\cdots \text{upto}\infty$
$\Rightarrow S(1-x)=\frac{1}{1-x}[\because \text{Common ratio}=x<1]$
$\Rightarrow S=\frac{1}{(1-x{)}^2}$
Hence, the required sum is, $\frac{1}{{(1-\frac{10}{11})}^2}=121$