The coefficients of $x^{n}$ in the polynomial $(x +^{n} C_{0}) (x + 3^{n} C_{1}) (x + 5^{n} C_{2}) . . . (x + {(2 n + 1)}^{n} C_{n})$ is

n 2^{n}

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n . 2^{n + 1}

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(n + 1) . 2^{n}

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n . 2^{n - 1}

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Solution

The correct option is C $(n + 1) . 2^{n}$
Given polynomial is
$(x +^{n} C_{0}) . (x + 3^{n} C_{1}) . (x + 5^{n} C_{2}) . . . (x + {(2 n + 1)}^{n} C_{n}) = x^{n + 1} + x^{n} {^{n} C_{0} + 3^{n} C_{1} + {5.}^{n} C_{2} + . . . {(2 n + 1)}^{n} C_{n}} + x^{n - 1} (. . .) + . . .$
$∴$ Coefficients of $x^{n}$ in the expansion is
$n \sum r = 0 {(2 r + 1)}^{n} C_{r} = n \sum r = 0 2 r^{n} C_{r} + n \sum r = 0^{n} C_{r} = 2 n \sum r = 0 r . \frac{n}{r} .^{n - 1} C_{r - 1} + 2^{n}$
$= 2 n n \sum r = 0^{n - 1} C_{r - 1} + 2^{n} = 2 n . 2^{n - 1} + 2^{n} = (n + 1) 2^{n}$

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Similar questions

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