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 \(\displaystyle\sum_{r=1}^{n} r^2\).
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- 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 \(\displaystyle\sum_{r=1}^{n} r^2\).
