Question

Let f(x) be a function satisfying $f\left(x\right)+f(x-1)=x^2\forall x\in R$. If f(0) =20, then remainder when f(50) is divided by 1000, is

Answer 1

$f\left(50\right)={50}^2-f\left(49\right)$
$={50}^2-({49}^2-f(49\left)\right)={50}^2-{49}^2+f\left(48\right)$
$f\left(50\right)={50}^2-{49}^2+f\left(48\right)$
$f\left(48\right)={48}^2-{47}^2+f\left(46\right)$
$f\left(46\right)={46}^2-{45}^2+f\left(44\right)$
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$f\left(2\right)=2^2-1^2+f\left(0\right)$
Adding the equations
$f\left(50\right)={50}^2-{49}^2+{48}^2-{47}^2+...+2^2-1^2+f\left(0\right))$
=(50+ ----+1) + f(0)
$=\frac{50}{2}\left(51\right)+20=1295$
So Remainder is 295