The coefficient of xn in expansion of (1+ x)(1- x)n is

Question

The coefficient of xn in expansion of (1+ x)(1- x)n is (A) (- 1)n-1 n  (B) (- 1)n (1 -n) (C) (-1)n-1 (n-1)2 (D) (n - 1 )

Tardigrade Admin · Accepted Answer

Coeff. of xn in (1+x)(1-x)n = coeff of xn in (1+x)(1 - nC1x + nC2x2 - ...+(-1)n × nCn xn) = (-1)n × nCn + (-1)n-1 nCn-1 = (-1)n + (-1)n-1 .n = (-1)n(1-n)

Q. The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is

$(- 1)^{n - 1} n$

$(- 1)^{n} (1 - n)$

$(- 1)^{n - 1} (n - 1)^{2}$

$(n - 1)$

Coeff. of $x^{n}$ in $(1 + x) (1 - x)^{n}$
= coeff of $x^{n}$ in
$(1 + x) (1 -^{n} C_{1} x +^{n} C_{2} x^{2} - ... + (- 1)^{n} \times^{n} C_{n} x^{n})$
$= (- 1)^{n} \times^{n} C_{n} + (- 1)^{n - 1}^{n} C_{n - 1} = (- 1)^{n} + (- 1)^{n - 1} . n$
$= (- 1)^{n} (1 - n)$

Q. The coefficient of xn in expansion of (1+x)(1−x)n is

(−1)n−1n

(−1)n(1−n)

(−1)n−1(n−1)2

(n−1)