2024 RI Promo Q5

1. Find \(\sum\limits_{r=0}^{n}{\left[ \left( n+2 \right)r+n \right]}\), giving your answer in terms of \(n\).[3]

[You may use the result \(\sum\limits_{r=1}^{n}{{{r}^{3}}=\frac{1}{4}{{n}^{2}}{{\left( n+1 \right)}^{2}}}\) for the rest of this question.]

2. By writing down the first two and the last two terms in the series, find \(\sum\limits_{r=1}^{n}{{{\left( r+2 \right)}^{3}}}\), giving your answer in terms of \(n\).[3]
3. Find \({{1}^{3}}-{{2}^{3}}+{{3}^{3}}-{{4}^{3}}+{{5}^{3}}-{{6}^{3}}+\ldots +{{\left( 2n-1 \right)}^{3}}-{{\left( 2n \right)}^{3}}\), giving your answer in terms of \(n\).[3]