Problem

(i) Prove by induction that for $n \in \mathbb{Z}^+$

$\left(\frac{5}{4}\right)^n - 1 = \frac{(2n + 3)n}{3 - 2n} \cdot \frac{1}{4^n}$

(ii) Prove by induction that for $n \in \mathbb{Z}^+$

$f(n) = 8^{2n + 1} + 6^{2n – 1}$

is divisible by $7$