Let a,b,cR.

Question

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 $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to:

Accepted Answer

$f(1)=a+b+c=3$. Setting $x=y=1$: $f(2)=2f(1)+1=7$. Setting $x=2,y=1$: $f(3)=f(2)+f(1)+2=12$.

In general $f(n+1)=f(n)+f(1)+n=f(n)+n+3$, so the partial sums $S_n=\sum_{k=1}^n f(k)$ satisfy the cubic-quadratic formula
$$S_n=\frac{n(n+1)(n+8)}{6}$$
(verified: $S_1=3,\ S_2=10,\ S_3=22$, matching $3,7,12$).

$$S_{10}=\frac{10\cdot11\cdot18}{6}=330.$$

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