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Fibonacci numbers with negative indices?

  1. Mar 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Let the Fibonacci sequence Fn be defined by its recurrence relation (1) Fn=F(n-1)+F(n-2) for n>=3. Show that there is a unique way to extend the definition of Fn to integers n<=0 such that (1) holds for all integers n, and obtain an explicit formula for the terms Fn with negative indices n.

    3. The attempt at a solution
    So I know the solution uses induction, and I think the first few negative terms should be F-1=-1, F-2=-1, F-3=-2 etc. So for the negative integers, Fn=F(n+1) + F(n+2) for n<0, but if the formula is supposed to extend to all integers n, that formula doesn't work...am I thinking about this problem wrong?
  2. jcsd
  3. Mar 17, 2010 #2
    Ok, so i worked on this a bit more, and found that the formula I'm trying to prove is Fn=F(n+2)-F(n+1), since this generates the negative terms...would the base case be n=1 and the induction hypothesis prove n=k-1?
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