Prove $\left(2017^{2018}+2017^{2017}\right)^{2018}>\left(2018^{2017}+2017^{2017}\right)^{2017}$ Can someone give me a hint for this problem (for secondary student): 

Prove that: $$\left(2017^{2018}+2017^{2017}\right)^{2018}>\left(2018^{2017}+2017^{2017}\right)^{2017}$$ 

P/s: I've thinking about using the fact that $n^{n+1}>(n+1)^n$ for $n$ is a natural number, but to prove this I've to use the derivative.

As the OP indicates, the desired inequality follows easily from the general inequality $(n+1)^n\lt n^{n+1}$, which holds for $n\ge3$ (but not for $n=1$ or $2$).  Here's the essential step in an easy induction proof.
Rewriting the inequality to prove as $\left( n+1\over n\right)^n\lt n$, let's assume by induction that we have
$$\left( n\over n-1\right)^{n-1}\lt n-1$$
Then, from ${n+1\over n}\lt {n\over n-1}$ (since $(n+1)(n-1)=n^2-1\lt n^2$), we have
$$\left( n+1\over n\right)^n\lt\left( n\over n-1\right)^n=\left( n\over n-1\right)\left( n\over n-1\right)^{n-1}\lt\left( n\over n-1\right)(n-1)=n$$
