8. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation
$$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$
- Find the general solution of this recurrence relation.
(5)
A particular solution of this recurrence relation has \(u _ { 0 } = 1\) and \(u _ { 1 } = k\), where \(k\) is a positive constant. All terms of the sequence are positive. - Determine the value of \(k\).
(3)