4 The sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation \(u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10\).
- Determine the general solution of the recurrence relation.
- Hence determine the particular solution of the recurrence relation for which \(u _ { 0 } = 6\) and \(u _ { 1 } = 10\).
- Show, by direct calculation, that your solution in part (b) gives the correct value for \(u _ { 2 }\). The sequence \(v _ { 0 } , v _ { 1 } , v _ { 2 } , \ldots\) is defined by \(v _ { n } = \frac { u _ { n } } { p ^ { n } }\) for some constant \(p\), where \(u _ { n }\) denotes the
particular solution found in part (b). particular solution found in part (b).
You are given that \(\mathrm { v } _ { \mathrm { n } }\) converges to a finite non-zero limit, \(q\), as \(n \rightarrow \infty\). - Determine \(p\) and \(q\).