- Sequences \(\left\{ x _ { n } \right\}\) and \(\left\{ y _ { n } \right\}\) for \(n \in \mathbb { N }\), are defined by
$$\begin{gathered}
x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n }
x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a
\end{gathered}$$
where \(a\) is a constant.
- Show that \(x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3\)
- Solve the second-order recurrence relation given in (a) to obtain an expression for \(x _ { n }\) in terms of \(a\) and \(n\).
Given that \(x _ { 7 } = 28225\)
- find the value of \(a\).