Answer
See below.
Work Step by Step
Proofs using mathematical induction consist of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number. Then we prove that then the statement also holds for $n + 1$.
Hence, here we have:
1) For $n=1: 5=5(3)^{1-1}$
2) Assume for $n=k: a_k=5(3)^{k-1}$. Then for $n=k+1$:
$a_{k+1}=3a_k=3(5(3)^{k-1})=5(3^k)$
Thus we proved what we wanted to.