Problem

9. (i) A sequence of numbers is defined by

$u_1 = 0 \quad u_2 = -6$

$u_{n+2} = 5u_{n+1} - 6u_n \quad n \geq 1$

Prove by induction that, for $n \in \mathbb{Z}^+$

$u_n = 3 \times 2^n - 2 \times 3^n$

(ii) Prove by induction that, for all positive integers $n$,

$f(n) = 3^{3n-2} + 2^{4n-1}$

is divisible by $11$