Induction Proofs - Mathematics

How do I show by mathematical induction that $2$ divides $n^2 - n$ for all $n$ belonging to the set of Natural Numbers

Answer

To prove this with induction (although there is a simpler way) you can proceed as follows.

For $n=1$ this is true since $2$ divides $0$. Let it be true for $n=k$ i.e. that

$$2 \underbrace{\mid}_{\text{ divides }} (k^2-k)=k(k-1)$$ then for $n=k+1$ you have that $$(k+1)^2-(k+1)=k^2+2k+\not1-k-\not1=k^2-k+2k=k(k-1)+2k$$ Now, observe that $2$ divides $k(k-1)$ by the induction hypothesis and obviously $2$ divides also $2k$. Thus $2$ divides $k(k-1)+2k$ and this completes the proof by induction.