I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no idea how induction worked, I showed him some typical examples.
Now he showed me an exercise he was having trouble with, which states that one should prove $$n^{n+1} > (n+1)^n$$ for all $n \geq 3$. I have to admit that I too have trouble showing this inequality, as in all of my attempts, my lower bound is too low. Also, I have not yet figured out, how one gets the $(n+2)^{n+1}$, primarely the number $2$ is a problem. I think this might be solved using the binomial theorem, however, I don't think they have already seen the binomial theorem in school.
Is there an easy method to show this inequality by induction not using the binomial theorem? If no: How can one show it using the binomial theorem?
Thanks for answers in advance.