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Evaluate the Following: N ∑ K = 1 ( 2 K + 3 K − 1 ) - Mathematics

Evaluate the following:

\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]

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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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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.3 | Q 3.2 | Page 28
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