# Reverse Induction?

by Char. Limit
Tags: induction, reverse
 PF Gold P: 1,930 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?
 Emeritus Sci Advisor Thanks PF Gold P: 15,876 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?
PF Gold
P: 1,930
 Quote by micromass 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?
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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