discrete mathematics - Proof by Induction: Solving $1+3+5+cdots+(2n-1)$

The question asks to verify that each equation is true for every positive integer n.

The question is as follows:

$$1+ 3 + 5 + \cdots + (2n - 1) = n^2$$

I have solved the base step which is where $n = 1$.

However now once I proceed to the inductive step, I get a little lost on where to go next:

Assuming that k is true (k = n), solve for k+1:

(2k - 1) + (2(k+1) - 1)

(2k - 1) + (2k+2 - 1)
(2k - 1) + (2k + 1)

This is where I am stuck. Do I factor these further to obtain a polynomial of some sort? Or am I missing something?

Answer

Assume true for $k$. Then consider the case $k+1$, you got $$1+3+\cdots+(2k-1)+(2(k+1)-1)$$ which is equal by inductive hypothesis $$k^2+(2k+1)=(k+1)^2$$ and this closes the induction.