2018 DHS Promo Q3
It is given that $\text{f}\left( r \right)=2{{r}^{3}}-3{{r}^{2}}+r-4$.
(i)
Show that $\text{f}\left( r+1 \right)-\text{f}\left( r \right)=k{{r}^{2}}$, where $k$ is a constant to be determined.
[2]
(ii)
Deduce a formula for $\sum\limits_{r=1}^{n}{{{r}^{2}}}$, fully factorising your answer.
[3]
(iii)
Hence show that $\sum\limits_{r=4}^{n-3}{{{\left( r-1 \right)}^{2}}=\frac{1}{6}\text{g}\left( n \right)-5}$, where $\text{g}\left( n \right)$ is to be determined.
[3]
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