$5$ Recurrence Relations

For each part, find the asymptotic order of growth of $T$; that is$,$ find a function $g$ such that $T(n)=(g(n)).$ Show your reasoning and do not directly apply the Master Theorem; doing so will yield $0$ credit.

In all subparts, you may ignore any issues arising from whether a number is an integer.

$($a$)T(n)=2T(\frac{n}{3})+5n$

$($b$)T(n)=169T(\frac{n}{170})+(n)$

$($c$)$ An algorithm A takes $(n^2)$ time to partition the input into $5$ sub$-$problems of size $\frac{n}{5}$ each and then recursively runs itself on $3$ of those subproblems. Describe the recurrence relation for the run$-$time $T(n)$ of A and find its asymptotic order of growth.

$2$