Answer
See below.
Work Step by Step
Proofs using mathematical induction consist 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 prove that the statement also holds for $n + 1$.
Hence here:
1) For $n=1: 2(1)-1=1^2$.
2) Assume for $n=k: 1+3+...+2k-1=k^2$. Then for $n=k+1$:
$1+3+...+2k-1+2k+1=k^2+2k+1=(k+1)^2.$
Thus we proved what we wanted to.
