Question

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

Answer 1

Answer: 2500

Let $l+1/50=x$ . Let S be the sum of $50$ terms of the given series. Then,
$S=1+2x+3x^2+4x^3+...+49x^{48}+5O{x}^{49}...(1)$
$\frac{xS=x+2x^2+3x^3+.....+49x^{49}+50x^{50}}{(1-x)S=1+x+x^2+x^3+...+x^{49}-50x^{50}}$
[Subtracting (2) from (1)]
$\Rightarrow S(1-x)=\frac{1-x^{50}}{1-x}-50x^{50}$
$\Rightarrow S(-1/50)=-50(1-x^{50})-50x^{50}$
$\Rightarrow \frac{1}{50}S=50$
$\Rightarrow S=2500$