Question

Find the sum of the series upto $n$ terms $1+\frac{1+2}{2}+\frac{1+2+3}{3}+\frac{1+2+3+4}{4}+$.....?

Answer 1

$1+\frac{1+2}{2}+\frac{1+2+3}{3}+........+\frac{1+2+3+...+n}{n}$

$i.e,\sum \frac{n(n+1)}{2n}$

$=\frac{1}{2}\sum (n+1)$

$=\frac{1}{2}(n\frac{(n+1)}{2}+n)$

$=\frac{n}{2}\frac{(n+1+2)}{2}$

$=\frac{n}{2}(n+3)$