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Functions: Recurrence Relation

XAT 2016 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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f is a function for which f(1)=1f(1) = 1 and f(x)=2x+f(x−1)f(x) = 2x + f(x - 1) for each natural number x≥2x \ge 2. Find f(31)f(31).

  1. A.

    869

  2. B.

    929

  3. C.

    951

  4. D.

    991

  5. E.

    None of the above

Answer

D

Explanation

f(1)=1f(1) = 1 f(2)=2(2)+1=5f(2) = 2(2) + 1 = 5 f(3)=2(3)+5=11f(3) = 2(3) + 5 = 11 f(4)=2(4)+11=19f(4) = 2(4) + 11 = 19 In general, f(x)=f(1)+2∑k=2xk=1+2(x(x+1)2−1)=x2+x−1f(x) = f(1) + 2\sum_{k=2}^x k = 1 + 2\left(\frac{x(x+1)}{2} - 1\right) = x^2 + x - 1. For x=31x = 31: f(31)=312+31−1=961+30=991f(31) = 31^2 + 31 - 1 = 961 + 30 = 991.

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