2021 MI P1 Q7

A sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\),…. is such that \({{u}_{n}}={{a}^{n+1}}-{{\left( n+1 \right)}^{a}}\), where \(a\) is a constant and \(n\ge 1\).

1. Given that \({{u}_{1}}=0\), find \({{u}_{3}}\).[2]

For the rest of this question, let \(a=2\). It is given that \(\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}\left( n+1 \right)\left( 2n+1 \right)}\).

2. Find \(\sum\limits_{r=1}^{n}{{{u}_{r}}}\) in terms of \(n\). (You need not simplify your answer.)[4]
3. Another sequence \({{v}_{1}}\), \({{v}_{2}}\), \({{v}_{3}}\),… is such that \(\sum\limits_{r=1}^{9}{\left( {{v}_{r+1}}-{{v}_{r}} \right)=\sum\limits_{r=1}^{10}{{{u}_{r}}}}\). Given that \({{v}_{1}}={{u}_{1}}\), find the value of \({{v}_{10}}\).[2]