RI Summation Tutorial Q6

For this question, you may use the results

\(\sum\limits_{r=1}^{n}{{{r}^{2}}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\) and \(\sum\limits_{r=1}^{n}{{{r}^{3}}}=\frac{{{n}^{2}}{{\left( n+1 \right)}^{2}}}{4}\).

1. Find \(\sum\limits_{r=1}^{n}{{{r}^{2}}\left( 2r-1 \right)}\) in terms of \(n\).[2]
2. Find \(\sum\limits_{r=1}^{n}{{{r}^{2}}\left( r-1 \right)}\) in terms of \(n\). Hence find \(\sum\limits_{r=2}^{n-1}{r{{\left( r+1 \right)}^{2}}}\) in terms of \(n\).[5]
3. Without using a graphing calculator, find the sum of the series
   \(4\left( 25 \right)-5\left( 36 \right)+6\left( 49 \right)-7\left( 64 \right)+\ldots -59\left( 3600 \right)\)[3]