Denoting the sum of $n$ odd numbers $1+3+5+\cdots+(2n-1) = 1+3+5+\cdots+(2n+1)$? Let's say we are denoting the sum of $n$ odd numbers. 
Then in symbols $1+3+5+\cdots+(2n-1)$.
If we substitute $(k+1)$ for $n$. $2n-1=2k+1$ 
So $1+3+5+...+2k+1$
Then can we use  $1+3+5+\cdots+(2n+1)$ instead of $1+3+5+\cdots+(2n-1)$?
Logically, I think they are the same, but when I think of the number of terms, $1+3+5+\cdots+(2n+1)$ has one more term than $1+3+5+\cdots+(2n-1)$.
I also think $1+3+5+\cdots+(2n+1)$ is equal to $1+3+5+\cdots+(2n-1)+(2n+1)$, so it would be contradiction. But I don't know how to explain $1+3+5+\cdots+(2n-1)=1+3+5+\cdots+2k+1$ is a contradiction.

Suppose your inductive hypothesis is $$1+3+5+...+(2n-1) = n^2.$$
You initially show the inductive hypothesis is true when $n=1$, e.g. by saying $1=1^2$ is indeed correct
Then you assume the inductive hypothesis is true for $n=k$ so $1+3+5+...+(2k-1) = k^2.$  You add $(2k+1)$ to both sides so the left hand side becomes $1+3+5+...+(2k+1)$ and the right hand side becomes $k^2 +2k +1 = (k+1)^2$.  Since these are equal, this shows the inductive hypothesis would then be true for $n=k+1$. 
So using the axiom of induction, this proves the inductive hypothesis.
There is no need for contradiction.
