Understanding this proof

Question

I don't know how they come from the step prior to the last to the last. If somebody could explain what happens there, that would be appreciated.

Thanks in advance!

Answer 1

It is an instance of the distributive property (Wikipedia link): $$\large a\cdot c\;+\;b\cdot c=(a+b)\cdot c$$ In your instance, we have $$\large a=n,\qquad b=2,\qquad c=\frac{n+1}{2}$$

Answer 2

$$\frac{n(n+1)}{2}+\frac{2(n+1)}{2}=\frac{n(n+1)+2(n+1)}{2}$$

Now make $n+1$ a common factor at the top.