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

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 $5.3^{n+1}+2n$?

Answer:

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

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