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 $n=1: 5^1-1=4$ is divisible by $4$.
2) Assume for $n=k: 5^k-1$ is divisible by $4$. Then for $n=k+1$:
$5^{k+1}-1=5^k\cdot5-1=5\cdot 5^k-5+4=5(5^k-1)+4$
Notice that $5^k-1$ is divisible by $4$ by the induction hypothesis above. Multiplying this number by $5$ will not change its divisibility by $4$. Similarly, adding $4$ will mean that it is still divisible by $4$. Thus, we proved what we had to.