Question

Write the sum of 20 terms of the series $1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+....$

Answer 1

Let the nth term be $a_n$.
Here,
$a_n=\frac{1}{n}\left(1+2+3+...+n\right)=\left(\frac{n+1}{2}\right)$
We know:
$S_n=\sum _{k=1}^n{a}_k$
Thus, we have:
$S_{20}=\sum _{k=1}^{20}{a}_k$
$$\begin{gathered} =\frac{1}{2}\left[\sum _{k=1}^{20}\left(k+1\right)\right] \\ =\frac{1}{2}\left[\sum _{k=1}^{20}k+20\right] \\ =\frac{1}{2}\left[\frac{20\left(21\right)}{2}+20\right] \\ =\frac{1}{2}\left[230\right] \\ =115 \end{gathered}$$