First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

$n \cdot 2^{n}$
B

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

$(n + 1) 2^{n}$
D

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

Solution

There are total (n + 1) factors, Iet P(x) = 0
$Let (x +^{n} C_{0}) (x + 3^{n} C_{1}) (x + 5^{n} C_{2}) \dots . . . . . (x + {(2 n + 1)}^{n} C_{n})$
$= a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$
clearly , $α_{n}$ = 1 and roots of the equation P(x) = 0 are
$-^{n} C_{0}, - 3^{n} C_{1}, \dots$
Sum of the roots = $- a_{n - 1} / a_{n} = -^{n} C_{0} - 3^{n} C_{1} - 5^{n} C_{2} \dots$
$\Rightarrow a_{n - 1} = (n + 1) 2^{n}$

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