Q:

Help with quadratic sequence for son's homework

I'm sorry this is really basic. I'm terrible with math and I'm struggling to help my son with homework:
Find an expression in terms of n for the nth term in this sequence:
$0 , 9 , 22 , 39, 60, \ldots$
We can get the $2n^{2}$ part of the answer but just can't get the rest. Any help gratefully appreciated.
Sorry again for there basic question.

A:

Given that the sequence is quadratic, start by assuming that
$$a_n=An^2+Bn+C$$
for some $A,B,C$. Since $a_0=0$, it follows immediately that $C=0$ and
$$a_n=An^2+Bn$$
Furthermore, since $a_1=9$ and $a_2=22$, you have
$$a_1=A+B=9$$
$$a_2=4A+2B=22$$
and you have the system of equations
$$A+B=9$$
$$4A+2B=22$$
When solved, this yields $A=2$ and $B=7$. Thus you have
$$a_n=2n^2+7n$$

