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AP EAMCET2005MathematicsBinomial Theorem

If |x| < 1 2 , then the coefficient of x^r in the expansion of 1+2 x (1-2 x)^2 , is

Options

  1. Ar 2^r
  2. B(2 r-1) 2^r
  3. Cr 2^ 2 r+1
  4. D(2 r+1) 2^r

Correct answer

D. (2 r+1) 2^r

Step-by-step solution

aligned & 1+2 x (1-2 x)^2 =(1+2 x)(1-2 x)⁻² &=(1+2 x) (1+ 2 1 ! (2 x)+ 23 2 ! + + 23 r (r-1) ! (2 x)^ r-1 . & .+ 2 3 4 (r+1)(2 x)^r r ! ) aligned The coefficient of x^r aligned & =2 r ! (r-1) ! 2^ r-1 + (r+1) ! r ! 2^r & =r 2^r+(r+1) 2^r=2^r(2 r+1) aligned

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