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 $n=1: -2=-0.5(1)(1+3)$.
2) Assume for $n=k: -2-3-...-(k+1)=-0.5k(k+3)$. Then for $n=k+1$:
$-2-3-...-(k+1)-(k+2)=-0.5k(k+3)-(k+2)=-0.5k^2-1.5k-k-2=-0.5(k+1)(k+4)=-0.5(k+1)((k+3)+1).$
Thus we proved what we wanted to.