Question

The sum to $50$ terms of the series $1+2(1+\frac{1}{50})+3{(1+\frac{1}{50})}^2+\dots$ is given by

• A. $2500$
• B. $2550$
• C. $2450$
• D. None of these

Answer 1

The correct option is A $2500$

Let $1+\frac{1}{50}=x$
Let $S$ be the sum of $50$ terms of the given series.
Then,
$S=1+2x+3x^2+4x^3+\dots +49x^{48}+50x^{49}...\left(1\right)$
$\underset{–}{xS=x+2x^2+3x^3+\dots +49x^{49}+50x^{50}}...\left(2\right)$
$(1-x)S=1+x+x^2+x^3+\dots +x^{49}-50x^{50}$

$\Rightarrow S(1-x)=\frac{1-x^{50}}{1-x}-50x^{50}$
$\Rightarrow S(-\frac{1}{50})=-50(1-x^{50})-50x^{50}$
$\Rightarrow \frac{S}{50}=50$
$\Rightarrow S=2500$