The value of $^{n} C_{0} + 3^{n} C_{1} + 5^{n} C_{2} + 7^{n} C_{3} + . . . . . . . + (2 n + 1)^{n} C_{1}$ equal to

2^{n}

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2^{n} + n {.2}^{n - 1}

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2^{n} (n + 1)

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Solution

The correct option is A $2^{n} (n + 1)$
Let $S =^{n} C_{0} + 3^{n} C_{1} + . . . . . + {(2 n + 1)}^{n} C_{n}$
$S = (^{n} C_{0} +^{n} C_{1} + . . . . +^{n} C_{n}) + 2 (^{n} C_{1} + 2^{n} C_{2} + 3^{n} C_{3} + . . . .)$
$S = {(1 + x)}^{n} + 2 \times \frac{d}{d x} {(1 + x)}^{n}$ at $x = 1$
$S = {(1 + 1)}^{n} + 2 \times n {(1 + 1)}^{n - 1} = 2^{n} + n 2^{n} = (n + 1) 2^{n}$

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