5. An increasing sequence \(\left\{ u _ { n } \right\}\) for \(n \in \mathbb { N }\) is such that the difference between the \(n\)th term of \(\left\{ u _ { n } \right\}\) and the mean of the previous two terms of \(\left\{ u _ { n } \right\}\) is always 6

1. Show that, for \(n \geqslant 3\) $$2 u _ { n } - u _ { n - 1 } - u _ { n - 2 } = 12$$ Given that \(u _ { 1 } = 2\) and \(u _ { 2 } = 8\)
2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
3. Show that as \(n \rightarrow \infty , u _ { n } \rightarrow k n\) where \(k\) is a constant to be determined. You must give reasons for your answer.