How to prove an inequality with a direct proof?

[Moodion]

Hello, I'm having trouble with an assigned problem, not really sure where to begin with it:

Prove that if \(\displaystyle a \in R\) and \(\displaystyle b \in R\) such that \(\displaystyle 0 < b < a\), then \(\displaystyle {a}^{n} - {b}^{n} \le {na}^{n-1}(a-b)\), where n is a positive integer, using a direct proof.

Pointers or the whole proof would be appreciated (might require some explanation afterwards!)

Thanks

 

[Opalg]

Hello, I'm having trouble with an assigned problem, not really sure where to begin with it:

Prove that if \(\displaystyle a \in R\) and \(\displaystyle b \in R\) such that \(\displaystyle 0 < b < a\), then \(\displaystyle {a}^{n} - {b}^{n} \le {na}^{n-1}(a-b)\), where n is a positive integer, using a direct proof.

Pointers or the whole proof would be appreciated (might require some explanation afterwards!)

Thanks

Hi Moodion and welcome to MHB!

Try factorising $a^n-b^n$ as $(a-b)(a^{n-1} + \ldots + b^{n-1})$ and then estimate the size of the second factor.

 

[Moodion]

Thanks for the hint, just what I needed