Proof by induction step $k(k+1)+2(k+1)=(k+1)(k+2)$

Question

I think I do understand the proof by induction right now but on the tutorial I am watching, I don't understand the last steep.

How is it possible to go from $k(k+1)+2(k+1)$ to $(k+1)(k+2)$?

thank you

Answer 1

Have a look at $k(k+1) + 2(k+1)$. This is a sum of two terms, where each term contains the factor $k+1$. You can thus factor out $k+1$.

Maybe we can set $x=k+1$ to make things clearer. Your expression becomes $kx + 2x$. Here you can factor out $x$ to get $x(k+2)$. But $x$ was $k+1$, so it is the same as $(k+1)(k+2)$.