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.