Prove that $n^{(n+1)} > (n+1)^{n}$ when $n\geq 3$ Firstly, for $n=3$,
$$3^4 > 4^3$$.
Secondly, $(n+1)^{(n+2)} > (n+2)^{(n+1)}$
Now I'm stuck. 

That is equivalent to proving that $a_n = \left(1+\frac{1}{n}\right)^n$ is an increasing sequence.
That follows, in a very straightforward way, from the AM-GM inequality, since:
$$\left(1+\frac{1}{n}\right)^n\cdot 1\leq\left(\frac{1+n\cdot\left(1+\frac{1}{n}\right)}{n+1}\right)^{n+1}=\left(1+\frac{1}{n+1}\right)^{n+1}. $$
