Answer
See below.
Work Step by Step
Proofs using mathematical induction consists 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:
1) For $n=1: 1=\frac{1}{4}(5^1-1)$.
2) Assume for $n=k: 1+5+...+5^{k-1}=\frac{1}{3}(5^k-1)$. Then for $n=k+1$:
$1+5+...+5^{k-1}+5^k=\frac{1}{4}(5^k-1)+5^k=\frac{5}{4}\cdot5^k-\frac{1}{4}=\frac{1}{4}(5^{k+1}-1).$
Thus we proved what we wanted to.