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Question
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
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Solution
\[S_{11} = \sum\nolimits_{n = 1}^{11} \left( 2 + 3^n \right)\]
\[ \Rightarrow S_{11} = \sum\nolimits_{n = 1}^{11} 2 + \sum\nolimits_{n = 1}^{11} 3^n \]
\[ \Rightarrow S_{11} = 2 \times 11 + \left( 3 + 3^2 + 3^3 + . . . + 3^{11} \right)\]
\[ = 22 + 3\left( \frac{3^{11} - 1}{3 - 1} \right) \]
\[ = 22 + \left( \frac{177147 - 1}{2} \right)\]
\[ = 22 + 265719 \]
\[ = 265741\]
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