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Math Help - Induction Help with inequality

  1. #1
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    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.
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  2. #2
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    Re: Induction Help with inequality

    Quote Originally Posted by eric34 View Post
    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.
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  3. #3
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    Re: Induction Help with inequality

    Indeed. The instructor presented it as he was reviewing induction, and offered a few points of "extra credit" for a correct proof.
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  4. #4
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    Re: Induction Help with inequality

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