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 $n=1: 1=0.5(3^1-1)$.
2) Assume for $n=k: 1+3+...+3^{k-1}=0.5(3^k-1)$. Then for $n=k+1$:
$1+3+...+3^{k-1}+3^k=0.5(3^k-1)+3^k=1.5\cdot3^k-0.5=0.5(3^{k+1}-1).$
Thus we proved what we wanted to.