The function f is given by
$$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$
Express
$$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$
in the form \(k \times 15 ^ { n }\).
Prove by induction that \(15 ^ { n } - 8 ^ { n - 2 }\) is a multiple of 7 for all integers \(n \geqslant 2\).