Question $9[4$ marks $+$ Bonus $5$ marks $($to an assignment maximum of $50$ marks$)]$

Let $f(n)$ be defined by

$f(1)=57$

$f(n)=3125f(\frac{n}{5})+71n^5$

if $n>1$ and $n=5^m,$ where m is a positive integer.

$($a$)$ By using the principles of Recurrence Relation, find a general formula

for $f(n)$

$($b$)$ Hence show that $f(n)=(n^{5}lo{g}_{5}n).$

$[$Bonus $-5]$