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 A $(n + 1) {.2}^{n}$
Given,
$(x + {^{n} C}_{0}) (x + 3 {^{n} C}_{1}) (x + 5 {^{n} C}_{2}) . . . (x + (2 n + 1) {^{n} C}_{n})$
$= x^{n + 1} + x^{n} [{^{n} C}_{0} + 3 {^{n} C}_{1} + 5 {^{n} C}_{2} + . . . . . (2 n + 1) {^{n} C}_{n}] + x^{n - 1} (. . .) . .$
Thus the coefficient of $x^{n}$ in the expression is
$n \sum r = 0 {(2 r + 1)}^{n} C_{r} = n \sum r = 0 {(2 r)}^{n} C_{r} + n \sum r = 0 {(1)}^{n} C_{r}$
$= 2 \sum_{r = 0}^{n} {(n)}^{n - 1} C_{r - 1} + 2^{n} = 2 n \sum_{r = 0}^{n}^{n - 1} C_{r - 1} + 2^{n} = 2 n . 2^{n - 1} + 2^{n}$

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