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.

Hint:
$$
\frac{2017^{2018}}{2017^{2017}}=2017
$$
but
$$
\begin{align}
\frac{2018^{2017}}{2017^{2017}}
&=\left(1+\frac1{2017}\right)^{2017}\\
&\le\left(1+\frac1{2017}\right)^{2018}\\
&\le\left(1+\frac1{2016}\right)^{2017}\\
&\qquad\ \ \vdots\\
&\le\left(1+\frac11\right)^2\\[6pt]
&=4
\end{align}
$$
To show the last series of inequalities, we can use Bernoulli's Inequality, which has a very simple inductive proof:
$$
\begin{align}
\frac{\left(1+\frac1{n-1}\right)^n}{\left(1+\frac1n\right)^{n+1}}
&=\frac{n-1}n\left(1+\frac1{n^2-1}\right)^{n+1}\\
&\ge\frac{n-1}n\left(1+\frac1{n-1}\right)\\[9pt]
&=1
\end{align}
$$
