- The complementary function for the second order recurrence relation
$$u _ { n + 2 } + \alpha u _ { n + 1 } + \beta u _ { n } = 20 ( - 3 ) ^ { n } \quad n \geqslant 0$$
is given by
$$u _ { n } = A ( 2 ) ^ { n } + B ( - 1 ) ^ { n }$$
where \(A\) and \(B\) are arbitrary non-zero constants.
- Find the value of \(\alpha\) and the value of \(\beta\).
Given that \(2 u _ { 0 } = u _ { 1 }\) and \(u _ { 4 } = 164\)
- find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
(6)