The coefficient of xn in the polynomial x+nC0x+3 nC1x+5 nC2…….x+(2n+1) nCn is

# The coefficient of xn in the polynomial  is

1. A

$\mathrm{n}\cdot {2}^{\mathrm{n}}$

2. B

$\mathrm{n}\cdot {2}^{\mathrm{n}+1}$

3. C

$\left(\mathrm{n}+1\right){2}^{\mathrm{n}}$

4. D

$\mathrm{n}\cdot {2}^{\mathrm{n}}+1$

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### Solution:

There are total (n + 1) factors, Iet P(x) = 0

$={\mathrm{a}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{x}}^{\mathrm{n}-1}+\dots +{\mathrm{a}}_{1}\mathrm{x}+{\mathrm{a}}_{0}$

clearly , ${\mathrm{\alpha }}_{\mathrm{n}}$ = 1 and roots of the equation P(x) = 0 are

${-}^{\mathrm{n}}{\mathrm{C}}_{0},-{3}^{\mathrm{n}}{\mathrm{C}}_{1},\dots$

Sum of the roots = $-{\mathrm{a}}_{\mathrm{n}-1}/{\mathrm{a}}_{\mathrm{n}}={-}^{\mathrm{n}}{\mathrm{C}}_{0}-{3}^{\mathrm{n}}{\mathrm{C}}_{1}-{5}^{\mathrm{n}}{\mathrm{C}}_{2}\cdots$

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