# What will be the sum of n terms of the series whose $n^{th}$ term is $5.3^{n+1}+2n$?

## Question:

What will be the sum of n terms of the series whose $n^{th}$ term is $5.3^{n+1}+2n$?

Here $a_n=5.3^{n+1}+2n$
We have have to find $s_n$.
$\therefore s_n=\displaystyle\sum_{k=1}^{n}a_k$
$\therefore s_n=\displaystyle\sum_{k=1}^{n}\left(5.3^{k+1}+2k\right)$
$\therefore s_n=\displaystyle\sum_{k=1}^{n}5.3.3^k+\displaystyle\sum_{k=1}^{n}2k$
$\therefore s_n=15\displaystyle\sum_{k=1}^{n}3^k+2\displaystyle\sum_{k=1}^{n}k$
$\therefore s_n=15[3\left(\frac{3^n-1}{3-1}\right)]+2[\frac{n(n+1)}{2}]$
$\therefore s_n=\frac{45}{2}(3^n-1)+n(n+1)$