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?
the first 8 square numbers represented as bars with a curve going through the top left corners of the bars

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