Determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 0\).
Using your answer to part (a), determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 20 n ^ { 2 } + 60 n\).
In the rest of this question the sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation in part (b). You are given that \(u _ { 0 } = - 9\) and \(u _ { 1 } = - 12\).
Determine the particular solution for \(\mathrm { u } _ { \mathrm { n } }\).
You are given that, as \(n\) increases, once the values of \(u _ { n }\) start to increase, then from that point onwards the sequence is an increasing sequence.
Use your answer to part (c) to determine, by direct calculation, the least value taken by terms in the sequence. You should show any values that you rely on in your argument.