Title4

Proving Odd

A generalization for all even numbers is 
$$
2k
$$

where k is an integer. Therefore, a generalization for all odd numbers would be. 

$$
2k+1
$$

To prove that squaring an odd integer is odd, let's square our expression. 

$$
\left( 2k + 1 \right)^{2} = 4k^2 + 4k + 1
$$

Let's factor out $4k$

$$
4k\left(k+1\right) + 1
$$

$$
2\left(k\left(k+1\right)\right) + 1
$$

Since $\left(k\left(k+1\right)\right)$ is still an integer, and we're working with generalizations, we can just merge that back into a generic constant $k$, and we're back to our original odd-form. 

$$
2k + 1
$$