This problem is a little more challenging. Most exercises in a class like this one can be done very quickly. However, the values of the techniques we learn in this and other basic classes lies in their being building blocks for the solutions of complicated problems. Here is a problem that requires quite a few steps, each of which is one that we have practiced in this class. But we have to put them all together to reach the final answer.
Let $$f(n) = 1^2 + 2^2 + 3^2 + \cdots + n^2.$$ Thus $$\begin{array}{rll} f(1) & = 1^2 &= 1, & \cr f(2) & = 1^2 + 2^2 &= 5, & \cr f(3) & = 1^2 + 2^2 + 3^2 &= 14, & \cr f(4) & = 1^2 + 2^2 + 3^2 +4^2 &= 30, & \cr \end{array}$$ etc.
It turns out that $f$ is a polynomial of degree 3 in $n$. Figure out the coefficients of $f$:
$f(n)=$ $n^3$ $+$ $n^2$ $+$ $n$ $+$ ,
One way to solve the problem is described in this
Hint: Write $$\begin{array}{rcl} f(n) &=& \phantom{+\quad} a \\ && +\quad b (n-1) \\ && +\quad c(n-1)(n-2) \\&& +\quad d(n-1)(n-2)(n-3)\\ \end{array}$$ where $a$, $b$, $c$, $d$ are constants that we still have to determine. The significance of writing $f$ in this form is that when $n=1$ only the first term is non-zero, when $n=2$ only the first two terms are non-zero, and so on. Note that $f(1) = a$, and so $a$ must be 1 (since the terms involving $b$, $c$, and $d$ are zero). Knowing $a$, ask what $f(2)$ is, use that value to figure out $b$, and go on from there. Once we know $a$, $b$, $c$ and $d$, we can convert $f$ to standard form.