2018 DHS Promo Q3

It is given that \(\mathrm{f}\left( r \right)=2{{r}^{3}}-3{{r}^{2}}+r-4\).

1. Show that \(\mathrm{f}\left( r+1 \right)-\mathrm{f}\left( r \right)=k{{r}^{2}}\), where \(k\) is a constant to be determined.[2]
2. Deduce a formula for \(\sum\limits_{r=1}^{n}{{{r}^{2}}}\), fully factorising your answer.[3]
3. Hence show that \(\sum\limits_{r=4}^{n-3}{{{\left( r-1 \right)}^{2}}=\frac{1}{6}\mathrm{g}\left( n \right)-5}\), where \(\mathrm{g}\left( n \right)\) is to be determined.[3]