Thread: Induction Help with inequality

Hello folks. I'm having a difficult time with the below induction problem, and I would greatly appreciate any nudge in the right direction.

a^n - b^n <= n*a^(n-1) *(a-b) for any n>1 and 0<b<a

I was able to prove my basis step at n=1

I am having difficulty with my inductive step of n+1.

After obtaining a^(n+1) - b^(n+1) <= n*a^(n+1) + a^(n+1) - n*b*a^n - b*a^n, I am at a loss as to how to obtain anything remotely intuitive/readable. I have tried manipulating factors and cancelling a variety of things, but have been unable to obtain anything much cleaner than this. Again, any thoughts would be appreciated.

Originally Posted by eric34

Hello folks. I'm having a difficult time with the below induction problem, and I would greatly appreciate any nudge in the right direction.
a^n - b^n <= n*a^(n-1) *(a-b) for any n>1 and 0<b<a

Must you do this using induction?

There is a much easier way to prove this with mean value theorem.

Indeed. The instructor presented it as he was reviewing induction, and offered a few points of "extra credit" for a correct proof.

Anyone else happen to have any thoughts how I may tackle this using induction?