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Question
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Solution
\[S_n = \sum^n_{k = 1} \left( 2^k + 3^{k - 1} \right)\]
\[ = \sum^n_{k = 1} 2^k + \sum^n_{k = 1} 3^{k - 1} \]
\[ = \left( 2 + 4 + 8 + . . . + 2^n \right) + \left( 1 + 3 + 9 + . . . + 3^n \right) \]
\[ = 2\left( \frac{2^n - 1}{2 - 1} \right) + 1\left( \frac{3^n - 1}{3 - 1} \right) \]
\[ = \frac{1}{2}\left( 2^{n + 2} - 4 + 3^n - 1 \right) \]
\[ = \frac{1}{2}\left( 2^{n + 2} + 3^n - 5 \right)\]
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