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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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