Reverse Induction?

  1. Char. Limit

    Char. Limit 1,986
    Gold Member

    1. The problem statement, all variables and given/known data
    Say I had a problem like this:

    Prove that the nth derivative of x*e^(x) is (x+n)*e^(x) for all integer n.

    Can I use reverse induction to prove for negative n? For example...

    Say I proved it for my base case, n=0. In this case, the proof is trivial.

    Then I prove that if the nth derivative is (x+n)e^(x), then the (n+1)th derivative is (x+n+1)e^(x). (I didn't provide the proof because there's a similar homework problem here, and the proof is easy anyway.

    Can I then use reverse induction to prove that if the nth derivative is (x+n)e^(x), then the (n-1)th derivative is (x+n-1)e^(x), thus extending this case to negative derivatives (i.e., integrals)?

    Am I even making sense?
     
  2. jcsd
  3. micromass

    micromass 18,536
    Staff Emeritus
    Science Advisor

    Hmm, I'm not sure if this is correct. You do have to use reverse induction though. But isn't it easier to show "if it holds for -n, then it holds for -n-1". Or is this what you meant?
     
  4. Char. Limit

    Char. Limit 1,986
    Gold Member

    Well, that would probably work too. EDIT: Since my base case is n=0, I don't see much of a difference.
     
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