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

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