Problem

(i) A sequence of numbers $u_1, u_2, u_3, \ldots$ is defined by

$u_{n+1} = \frac{1}{3}(2u_n - 1) \quad u_1 = 1$

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

$u_n = 3\left(\frac{2}{3}\right)^n - 1$

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

Prove by induction that, for $n \in \mathbb{Z}^+$, $f(n)$ is a multiple of $7$