<s>[INST] Why is $\;n^2-\frac{n^2}{2} =\frac{n^2}{2}\;$? Could someone please expand on how to get from $\;\displaystyle\left( n^2-\frac{n^2}{2}\right)\;$ to $\;\left(\dfrac{n^2}{2}\right)\;?\;$ 
I can't seem to wrap my head around that. 
 [/INST] We want to prove that $n^2 - \frac{n^2}{2} = \frac{n^2}{2}$. In order to better understand what we want to do, let us simplify this equation a little bit by adding $\frac{n^2}{2}$ to both sides. We get $n^2 = \frac{n^2}{2} + \frac{n^2}{2}$. So in order to solve the problem, we just need to show that $\frac{n^2}{2} + \frac{n^2}{2} = n^2$. In order to show that these two things are equal, let's start with the left hand side and show that it equal to the right hand side. Since we know how to add fractions with the same denominator, we get 
$$\frac{n^2}{2} + \frac{n^2}{2} = \frac{2n^2}{2} = \frac{2}{2}n^2= n^2$$
Now we're done!
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