Question

Find sum of the series $S=1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3+4)+\dots ..$ upto $20$ terms.

Answer 1

Answer: 115.00

$S=1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3+4)+\dots \dots \dots$
General term, $T_n=\frac{1}{n}(1+2+3+\dots \dots \dots \dots \dots n)$
$T_n=\frac{1}{n}\left\{\frac{n(n+1)}{2}\right\}=\frac{1}{2}(n+1)$
$S_n=\frac{1}{2}\{\Sigma n+\Sigma 1\}=\frac{1}{2}\left\{\frac{n(n+1)}{2}+n\right\}$
$S_{20}=\frac{1}{2}\left[\frac{20\times 21}{2}+20\right]=115$