Question

Let
$f(x) = 2x^2 - x - 1\ and\ S = \\{ n \in \mathbb{Z} : |f(n)| \leq 800 \\}$
Then, the value of ∑n∈S f(n) is equal to ________.

Answer 1

The correct answer is 10620
$∵ |f(n)|≤800$
$⇒ -800 ≤ 2n^2-n-1≤800$
$⇒ 2n^2-n-801≤0$
$∴ n∈[\frac{-\sqrt{6409}+1}{4}, \frac{\sqrt{6409}+1}{4}]$ and $n∈z$
$∴ n = -19, -18, -17,.....,19,20.$
$∴ ∑(2x^2-x-1) = 2∑x^2-∑x-∑1$
$= 2.2.(1^2+2^2+...+19^2)+2.20^2-20-40$
= 10620