The coefficient of x - n in 1 + x n 1 + 1 x n is

1 + x n 1 + 1 x n = 1 + x 1 + 1 x n 1 + x n 1 + 1 x n = 1 + x + 1 x + 1 n 1 + x n 1 + 1 x n = 2 + x + 1 x n 1 + x n 1 + 1 x n = C 0 n 2 + x n · 1 x + C 1 n 2 + x n - 1 · 1 x 1 + … + C n n · 1 x n ∴ Coefficient of x - n = C n n = 1

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MathematicsBinomial TheoremGeneralMedium2 minPYQ_2018

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The coefficient of $x^{- n}$ in ${(1 + x)}^{n} {(1 + \frac{1}{x})}^{n}$ is

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Answer:

Solution:

${(1 + x)}^{n} {(1 + \frac{1}{x})}^{n} = {[(1 + x) (1 + \frac{1}{x})]}^{n}$

${(1 + x)}^{n} {(1 + \frac{1}{x})}^{n} = {[1 + x + \frac{1}{x} + 1]}^{n}$

${(1 + x)}^{n} {(1 + \frac{1}{x})}^{n} = {[(2 + x) + \frac{1}{x}]}^{n}$

${(1 + x)}^{n} {(1 + \frac{1}{x})}^{n} = {}^{n}C_{0} {(2 + x)}^{n} \cdot (\frac{1}{x}) + {}^{n}C_{1} {(2 + x)}^{n - 1} \cdot {(\frac{1}{x})}^{1} + \dots + {}^{n}C_{n} \cdot \frac{1}{x^{n}}$

$∴$ Coefficient of $x^{- n} = {}^{n}C_{n} = 1$

Stream:BITSATSubject:MathematicsTopic:Binomial TheoremSubtopic:General

⏱ 2mℹ️ Source: PYQ_2018

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