Evaluate the following: _ n=1 ^ 11 (2+3^n) - Vidyadip

Question

Evaluate the following:$\sum_\limits{\text{n}=1}^{11}(2+3^\text{n})$

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Answer

$\sum_\limits{\text{n}=1}^{11}(2+3^\text{n})$ $=(2+3^1)+(2+3^2)+(2+3^3)+\ \dots\ +(2+3^{11})$ $=2\times11+3^1+3^2+3^3+\dots+3^{11}$ $=22+\frac{3(3^{11}-1)}{(3-1)}$ $=22+\frac{3(3^{11}-1)}{2}$ $=\frac{44+3(177147-1)}{2}$ $=\frac{44+3(177146)}{2}$ $=265741$ So, $\sum_\limits{\text{n}=1}^{11}(2+3^\text{n})=265741$

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