3 A sequence is defined by the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 5 u _ { n }\) for \(n \geqslant 0\), with \(u _ { 0 } = 0\) and \(u _ { 1 } = 1\).
- Find an exact real expression for \(u _ { n }\) in terms of \(n\) and \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\).
A sequence is defined by \(v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }\) for \(n \geqslant 0\), where \(a\) is a positive constant.
- In each of the following cases, describe the behaviour of \(v _ { n }\) as \(n \rightarrow \infty\).
- \(a = 0.1\)
- \(a = 0.2\)
- \(a = 1\)