Review question

# Find an expression for the sum of $r^2$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6143

## Question

It is given that $\sum\limits_{r=-1}^n r^2$ can be written in the form $pn^3+qn^2+rn+s$, where $p,q,r$ and $s$ are numbers. By setting $n=-1,0,1$ and $2$, obtain four equations that must be satisfied by $p,q,r$ and $s$ and hence show that $\sum_{r=0}^n r^2=\frac{1}{6}n(n+1)(2n+1).$ Given that $\sum\limits_{r=-2}^n r^3$ can be written in the form $an^4+bn^3+cn^2+dn+e$, show similarly that $\sum_{r=0}^n r^3=\frac{1}{4}n^2(n+1)^2.$