Suppose we have to prove that $1+3+5+...(2n-1)=n^2$
Here is my "proof," which I did by induction. Since I am learning induction, I wanted to know whether I did it correctly (I don't think I did).
For the base case, suppose n=1. Then $2(1)-1=1^2$, which is clearly true.
Now suppose $n=k$, then $1+3+5+...+ (2k-1)=k^2$
For $n=k+1$, $1+3+5+...+(2k+1)=(k+1)^2$. Then $1+3+5+...+(2k+1)=k^2+2k+1$. The 2k+1's cancel out, leaving $1+3+5+...+2k-1=k^2$, which is true by the induction hypothesis. QED?