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Sum of Infinite Series

CAT 2002 Slot 1 · Quantitative Ability · Hard · Algebra

This is a hard Quantitative Ability question from the CAT 2002 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Let SS denote the infinite sum 2+5x+9x2+14x3+20x4+…2 + 5x + 9x^2 + 14x^3 + 20x^4 + \dots, where ∣x∣<1|x| < 1 and the coefficient of xn−1x^{n-1} is 12n(n+3),(n=1,2,… )\frac{1}{2}n(n+3), (n=1, 2, \dots). Then SS equals:

  1. A.

    \frac{2-x}{(1-x)^3}

  2. B.

    \frac{2-x}{(1+x)^3}

  3. C.

    \frac{2+x}{(1-x)^3}

  4. D.

    \frac{2+x}{(1+x)^3}

Answer

A

Explanation

S=2+5x+9x2+14x3+…S = 2 + 5x + 9x^2 + 14x^3 + \dots xS=2x+5x2+9x3+…xS = 2x + 5x^2 + 9x^3 + \dots S(1−x)=2+3x+4x2+5x3+…S(1-x) = 2 + 3x + 4x^2 + 5x^3 + \dots xS(1−x)=2x+3x2+4x3+…xS(1-x) = 2x + 3x^2 + 4x^3 + \dots S(1−x)2=2+x+x2+x3+⋯=2+x1−x=2−x1−xS(1-x)^2 = 2 + x + x^2 + x^3 + \dots = 2 + \frac{x}{1-x} = \frac{2-x}{1-x}. Therefore, S=2−x(1−x)3S = \frac{2-x}{(1-x)^3}.

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