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AP EAMCET202421 May 2024Evening ShiftMathematicsBinomial TheoremActual

In the expansion of 2 x+1 (1+x)(1-2 x) , the sum of the coefficients of the first 5 odd powers of x is

Options

  1. A5 3 + 8 9 (4^5-1 )
  2. B5 3 + 8 3 (4^5-1 )
  3. C- 5 3 + 8 9 (4^5-1 )
  4. D5 3 + 8 12 (4^5+1 )

Correct answer

A. 5 3 + 8 9 (4^5-1 )

Step-by-step solution

aligned & Let 2 x+1 (1+x)(1-2 x) & = A 1+x + B 1-2 x = A(1-2 x)+B(1+x) (1+x)(1-2 x) & 2 x+1=A(1-2 x)+B(1+x) aligned On comparing both sides, we get aligned & A=- 1 3 and B= 4 3 & 2 x+1 (1+x)(1-2 x) = -1 3 (1+x)⁻¹+ 4 3 (1-2 x)⁻¹ & = -1 3 [1-x+x^2- ]+ 4 3 [1+2 x+2^2 x^2+2^3 x^3+ ] aligned Sum of the coefficient of the first 5 odd power of x aligned & = -1 3 [-1-1-1-1-1]+ 4 3 [2+2^3+2^5+2^7+2^9 ] & = 5 3 + 4 3 [ 2 (4^5-1 ) 4-1 ]= 5 3 + 8 9 (4^5-1 ) aligned

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