If I=a^{2}+b^{2}+c^{2}, where a and b are consecutive integers and c=ab, then I is

  1. A. an even number and it is not a square of an integer
  2. B. an odd number and it is not a square of an integer
  3. C. square of an even integer
  4. D. square of an odd integer ✓

Correct Answer: D. square of an odd integer

Explanation

Since b = a+1, we have c = a(a+1). Substituting, I = a^2 + (a+1)^2 + a^2(a+1)^2 = a^4 + 2a^3 + 3a^2 + 2a + 1 = (a^2 + a + 1)^2. Since a(a+1) is always even, a^2 + a + 1 = a(a+1) + 1 is always odd, making I the square of an odd integer.

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