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 $n=1: 3=0.5(1)(1+5)$.
2) Assume for $n=k: 3+4+...+(k+2)=0.5k(k+5)$. Then for $n=k+1$:
$3+4+...+(k+2)+(k+3)=0.5k(k+5)+(k+3)=0.5k^2+2.5k+k+3=0.5(k+1)(k+6)=0.5(k+1)((k+5)+1).$
Thus we proved what we wanted to.