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CAT 2025 Slot 2 · Quant · Functions · hard

A function f satisfies f(x+y) = f(x) + f(y) + xy for all real x, y, and f(1) = 2. Find f(10).

Single correct option

A function f satisfies f(x+y) = f(x) + f(y) + xy for all real x, y, and f(1) = 2. Find f(10).
  1. A. 55
  2. B. 60
  3. C. 65
  4. D. 70
Show answer and solution

Answer: (C) 65

Put y = 1: f(n+1) = f(n) + f(1) + n = f(n) + n + 2. Starting f(1)=2: f(2)=2+1+2=5, f(3)=5+2+2=9, f(4)=9+3+2=14, and so on up to f(10). Following the pattern f(n) = f(1)·n + n(n−1)/2 = 2n + n(n−1)/2, at n=10: f(10) = 20 + 45 = 65.
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