Problem

In the Warm-up we saw that $n^2$ and $n^3$ can be used as lower and upper bounds for the value of $\displaystyle\sum_{r=1}^{n} r^2$: $n^2 ≤ \displaystyle\sum_{r=1}^{n} r^2 ≤ n^3$

We can use the diagrams below to find a different way to bound $\displaystyle\sum_{r=1}^{n} r^2$.

• Which of the areas represented in the two diagrams is bigger?
• Does this result generalise?

Use this information to provide either an upper or lower bound for $\displaystyle\sum_{r=1}^{n} r^2$.

• Can you find a function which bounds the graphical representation of $\displaystyle\sum_{r=1}^{n} r^2$ on the other side?

Use this function to provide a second bound for $\displaystyle\sum_{r=1}^{n} r^2$.