JEE Main 2017 (Offline)Question 1
Let $a,b,c\in\mathbb R$. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy$, $\forall x,y\in\mathbb R$, then $\sum_{n=1}^{10}f(n)$ is equal to :
Functions
Solution
$f(0)=0\Rightarrow c=0$; comparing $x^2$ terms $2a=1$, so $a=\frac12$, and $a+b=3$ gives $b=\frac52$. $\sum f(n)=\frac12\cdot385+\frac52\cdot55=330$.