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AP EAMCET202422 May 2024Morning ShiftMathematicsBinomial TheoremActual

If |x| 1 , then the number of terms in the expansion of [ 1 2 (1.2+2.3 x+3.4 x^2+ . . ) ]⁻²⁵ is

Options

  1. AInfinite
  2. B101
  3. C76
  4. D51

Correct answer

C. 76

Step-by-step solution

(1-x)⁻¹=1+x+x^2+x^3+x^4+ Differentiate w.r.t. x(1-x)⁻²=1+2 x+3 x^2+4 x^3+ Differentiate again w.r.t. x aligned & 2(1-x)⁻²=1 2+2 3 x+3 4 x^2+ & (1-x)⁻³= 1 2 (1 2+2 3 x+3 4 x^2+ ) aligned Now, [ 1 2 (1 2+2 3 x+3 4 x^2+ ) ]⁻²⁵= [(1-x)⁻³ ]⁻²⁵=(1-x)⁷⁵ Number of terms =75+1=76 .

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