2021 DHS P1 Q7

A sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), … is such that \({{u}_{n+2}}+a{{u}_{n+1}}+{{u}_{n}}=b\), where \(a\), \(b\) are constants and \(n\ge 1\).

1. Given that \({{u}_{1}}=c\), \({{u}_{2}}=0.5\), \({{u}_{3}}=5\), \({{u}_{4}}=14.5\) and \({{u}_{5}}=29\), where \(c\) is a constant, find \(a\), \(b\) and \(c\).[2]
2. By using the substitution \({{v}_{n}}={{u}_{n+1}}-{{u}_{n}}\), explain why the sequence \(\left\{ {{v}_{n}} \right\}\) is an arithmetic progression.[2]
3. By considering \(\sum\limits_{r=1}^{n-1}{{{v}_{r}}=\sum\limits_{r=1}^{n-1}{\left( {{u}_{r+1}}-{{u}_{r}} \right)}}\), find \({{u}_{n}}\) in terms of \(n\).[4]
4. Describe the behaviour of the sequence \(\left\{ {{u}_{n}} \right\}\) for large values of \(n\).[1]