Prove that the coefficient of $x^{n}$ in the expansion $(1 + x) (1 - x)^{n}$ is $(- 1)^{n} (1 - n)$ .

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Solution

$(1 + x) (1 - x)^{n}$
To prove :- Coefficient of $x^{n} = (- 1) [1 - n]$
$(1 + x)^{n} (1 - x)^{n} = (1 + x) [^{n} C_{0} +^{n} C_{1} (- 1)^{1} x +^{n} C_{2} (- x)^{2} . . . . . .^{n} C_{3} (- x)^{3} +^{n} C_{n - 1} (- x)^{n - 1} +^{n} C_{n} (- x)^{n}]$
Coefficient of $x^{n} =^{n} C_{n - 1} (- 1)^{n - 1} +^{n} C_{n} (- 1)^{n}$
$= (- 1)^{n} [^{n} C_{n} -^{n} C_{n - 1}]$
$= (- 1)^{n} [1 - \frac{n!}{(n - 1)!}]$
$= (- 1)^{n} [1 - \frac{n (n - 1)!}{(n - 1) 1}]$
Coefficient of $x^{n} = (- 1)^{n} [1 - n]$

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