Prove inequality $(n-1)\cdot a^n +b^n \geq n\cdot a^{n-1}\cdot b$ by induction

by Matthias W   Last Updated August 14, 2019 09:20 AM - source

let $a,b \in \mathbb{R}; a,b>0$
I have to prove the following inequality by induction:

$(n-1)\cdot a^n +b^n \geq n\cdot a^{n-1}\cdot b$ with $n\in \mathbb{N}\setminus{\{1}\}$

I was not able to use the induction hypothesis for hours and of course did not reach the desired result: $n\cdot a^{n+1} +b^{n+1} \geq (n+1)\cdot a^n\cdot b$.

Than I rearranged the inequality to
$(n-1)+(\frac{b}{a})^n \geq n\cdot \frac{b}{a}$ by dividing both sides by $a^n$

Doing induction for the rearranged inequality

Base case: assumption is true for $n=2$
$(2-1)+(\frac{b}{a})^2 \geq 2\cdot \frac{b}{a} \Leftrightarrow a^2+b^2\geq 2ab \Leftrightarrow (a-b)^2\geq 0$

It remains to prove that the assumptions holds for $n+1$
This means $n\cdot a^{n+1} +b^{n+1} \overset{?}{\geq} (n+1)\cdot a^n\cdot b$

Inductive step:
$(n+1-1)+(\frac{b}{a})^{n+1} \geq (n+1)\cdot \frac{b}{a}$
$\Leftrightarrow n+(\frac{b}{a})^n\cdot(\frac{b}{a}) \geq n\cdot (\frac{b}{a})+(\frac{b}{a})$
$\Leftrightarrow \frac{n\cdot a}{b}+(\frac{b}{a})^n\geq n+1$
$\Leftrightarrow \frac{n\cdot a\cdot a^n+b^n\cdot b}{b\cdot a^n}\geq n+1$
$\Leftrightarrow \frac{n\cdot a^{n+1}+b^{n+1}}{b\cdot a^n}\geq n+1$
$\Leftrightarrow n\cdot a^{n+1}+b^{n+1}\geq (n+1)\cdot a^n\cdot b$

So I reached the desired result, but did not use the induction hypothesis.

Can someone please point out the mistake(s) I have done?

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