Complex Numbers And Quadratic Equations

Question
CBSEENMA11015668

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

  • (n-1)

  • (-1)n(1-n)

  • (-1)n-1(n-1)2

  • (-1)n-1n

Solution

B.

(-1)n(1-n)

The coefficient of xn in expansion of (1+x)(1-x)n is = coefficient of xn + coefficient of xn-1
left parenthesis negative 1 right parenthesis to the power of straight n space fraction numerator straight n factorial over denominator straight n factorial 0 factorial end fraction space minus fraction numerator straight n factorial over denominator 1 factorial left parenthesis straight n minus 1 right parenthesis factorial end fraction
space equals space left parenthesis negative 1 right parenthesis to the power of straight n space open square brackets fraction numerator straight n factorial over denominator straight n factorial 0 factorial end fraction minus fraction numerator straight n factorial over denominator 1 factorial left parenthesis straight n minus 1 right parenthesis factorial end fraction close square brackets
equals space minus space left parenthesis 1 right parenthesis to the power of straight n space left square bracket 1 minus straight n right square bracket

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