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