- A recurrence system is defined by
$$\begin{aligned}
u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1
u _ { 1 } & = - 3 , u _ { 2 } = 18
\end{aligned}$$
Prove by induction that, for \(n \in \mathbb { N }\),
$$u _ { n } = ( - 3 ) ^ { n } n !$$