2021 ASRJC P1 Q8

By using method of difference, show that \(\sum\limits_{r=1}^{n}{\left( {{\left( r+1 \right)}^{3}}-{{r}^{3}} \right)={{\left( n+1 \right)}^{3}}-1}\).[2]

By simplifying \({{\left( r+1 \right)}^{3}}-{{r}^{3}}\) or otherwise, show that \(\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}\left( n+1 \right)\left( 2n+1 \right)}\).[4]

Hence find \(\sum\limits_{r=5}^{N}{{{\left( r+2 \right)}^{2}}}\), leaving your answer in terms of \(N\).[3]