# Write the sum of 20 terms of the series

Question:

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

Solution:

Let the $n$th term be $a_{n}$.

Here,

$a_{n}=\frac{1}{n}(1+2+3+\ldots+n)=\left(\frac{n+1}{2}\right)$

We know:

$S_{n}=\sum_{k=1}^{n} a_{k}$

Thus, we have:

$S_{20}=\sum_{k=1}^{20} a_{k}$

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

$=\frac{1}{2}\left[\sum_{k=1}^{20} k+20\right]$

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

$=\frac{1}{2}[230]$

$=115$