The coefficient of xn in x(x−a)(x−b) is

=(aa−b)1x−a+b(b−a)⋅1x−b=−(1−xa)−1a−b−(1−xb)−1b−aCoefficient of xn=−1(a−b)(1a)n−1(b−a)(1b)n

The coefficient of xn in x(x−a)(x−b) is

=(aa−b)1x−a+b(b−a)⋅1x−b=−(1−xa)−1a−b−(1−xb)−1b−aCoefficient of xn=−1(a−b)(1a)n−1(b−a)(1b)n

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The coefficient of $x^{n}$ in $\frac{x}{(x - a) (x - b)}$ is

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$\frac{a^{n} + b^{n}}{a - b} \frac{1}{a^{n} b^{n}}$

$\frac{a^{n} - b^{n}}{a - b} \frac{1}{a^{n} b^{n}}$

$\frac{a^{n} - b^{n}}{a + b} \frac{1}{a^{n} b^{n}}$

$\frac{a^{n} - b^{n}}{a - b} a^{n} b^{n}$

answer is B.

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Detailed Solution

$= (\frac{a}{a - b}) \frac{1}{x - a} + \frac{b}{(b - a)} \cdot \frac{1}{x - b}$

$= \frac{- {(1 - \frac{x}{a})}^{- 1}}{a - b} - \frac{{(1 - \frac{x}{b})}^{- 1}}{b - a}$

Coefficient of $x^{n} = - \frac{1}{(a - b)} {(\frac{1}{a})}^{n} - \frac{1}{(b - a)} {(\frac{1}{b})}^{n}$

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The coefficient of xn in x(x−a)(x−b) is

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The coefficient of $x^{n}$ in $\frac{x}{(x - a) (x - b)}$ is