For all $a,b,n \in \mathbb N$, $0 \leq n(a+b+1)-n^2 +b$

Question

I claim the following:

For all $a,b,n \in \mathbb N$, we have that $0 \leq n(a+b+1)-n^2 +b$.

This seems true for me... Although how do I really check if this is true? I was thinking about doing it by induction, but the fact that we have to deal with three variables is confusing me. How can I do this?

Thank you!

Answer 1

This is untrue. For $a=b=1$ and $n=100,$ we get $$n(a+b+1)-n^2+b=100(1+1+1)-100^2+1=-9699.$$

General advice: higher order terms dominate.