Question

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

Answer 1

$S=1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3 +4)+\ldots \ldots \ldots$
General term, $T_{n}=\frac{1}{n}(1+2+3 +\ldots \ldots \ldots \ldots \ldots n)$
$T _{ n }=\frac{1}{ n }\left\{\frac{ n ( n +1)}{2}\right\}=\frac{1}{2}( n +1)$
$S _{ n }=\frac{1}{2}\{\Sigma n +\Sigma 1\}=\frac{1}{2}\left\{\frac{ n ( n +1)}{2}+ n \right\}$
$S _{20}=\frac{1}{2}\left[\frac{20 \times 21}{2}+20\right]=115$