JEE Main 2022 (Online) 27th July Morning ShiftQuestion 1
Let $f(x)=2x^2-x-1$ and $S=\{n\in\mathbb{Z}:|f(n)|\le800\}$. Then, the value of $\sum_{n\in S}f(n)$ is equal to ________.
Quadratic Equation · Sign, Range and Inequalities of Quadratic Expressions
Answer$10620$
Solution
Concept. $f(n)\ge-1$ for all integers, so only $f(n)\le800$ matters; then add with standard sums. Formulas used. (1) $\sum_{k=1}^{m}k^2=\dfrac{m(m+1)(2m+1)}{6}$Step 1 - Range. $2n^2-n-801\le0$ has roots $\frac{1\pm\sqrt{6409}}{4}\approx-19.8,\ 20.3$. So $S=\{-19,\dots,20\}$ (40 integers). Step 2 - Sums. $\sum n^2=2\cdot\frac{19\cdot20\cdot39}{6}+400=5340$ and $\sum n=20$. Step 3 - Total. $$\sum(2n^2-n-1)=10680-20-40=10620.$$ Answer: 10620