- (i) A sequence of numbers is defined by
$$\begin{gathered}
u _ { 1 } = 3
u _ { n + 1 } = 2 u _ { n } - 2 ^ { n + 1 } \quad n \geqslant 1
\end{gathered}$$
Prove by induction that, for \(n \in \mathbb { N }\)
$$u _ { n } = 5 \times 2 ^ { n - 1 } - n \times 2 ^ { n }$$
(ii) Prove by induction that, for \(n \in \mathbb { N }\)
$$f ( n ) = 5 ^ { n + 2 } - 4 n - 9$$
is divisible by 16