Let `f(x)=A x^2+B x+c ,w h e r eA ,B ,C` are real numbers. Prove that if `f(x)` is an integer whenever `x` is an integer, then the numbers `2A ,A+B ,a n dC` are all integer. Conversely, prove that if the number `2A ,A+B ,a n dC` are all integers, then `f(x)` is an integer whenever `x` is integer.

Explanation

Let us consider the integral values of x as 0, 1 , -1, then f(0), f(1) and f(-) are all integers. This means that C, A+ B + C and A - B + C are all integers . Therefore, C is integer and hence, A + B and A-B are also integers . Now, `2A = (A +B)+(A -B)`
Thus, `2A, A + B` and C are all integers . Conversely, let n `in` I. Then
`f(n) = An^(2) + Bn + C = 2 A [(n(n-1))/(2)]+ (A +B) n+C`
Now, A, A + B and C are all integers.
Also, `(n(n-1))/(2) = (Even)/(2)` = integer
Therefore, f(n) is an integer.