Question

$1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3+4)+.....$ upto $20$ terms is

• A. $110$
• B. $111$
• C. $115$
• D. $116$

Answer 1

The correct option is C $115$

Given

$S=1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+.....+\frac{1}{20}(1+2+3+....+20)$

$=\sum _{n=1}^{20}\frac{1}{n}\sum n$

$\sum _{n=1}^{20}\frac{1}{n}.\frac{n(n+1)}{2}=\frac{1}{2}\sum _{n=1}^{20}(n+1)$

$\frac{1}{2}{[\frac{n(n+1)}{2}+n]}_0^{20}$

$=\frac{1}{2}[\frac{20\left(21\right)}{2}+20]$

$=5\left(21\right)+10=115$