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Help with Fibonacci Transformation

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data
    A more efficient algorithm to calculate Fibonacci numbers applies the simultaneous transformation:

    T(a; b) = (a+b; a)

    repeatedly with a = 1 and b = 0 as initial values.

    What Fibonacci numbers result from T^k(1; 0)? Justify your answer (e.g., as proof by induction in k would be nice).


    2. Relevant equations



    3. The attempt at a solution

    Let k = 1

    T^1(1; 0) = (1; 1) = (F2; F1)

    Assume T^k(1; 0) = (Fk+1; Fk) and show T^(k+1)(1; 0) = (Fk+2; Fk+1)

    T^(k+1)(1; 0) = T^k(T^1(1; 0)) = T^k(1; 1) = (Fk+2; Fk+1)

    Is this enough to conclude my solution and justify the proof? Any help will be greatly appreciated.

    Thanks
     
  2. jcsd
  3. Oct 27, 2012 #2

    haruspex

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    I see no justification for that step. Your inductive hypothesis concerned T^k(1; 0), so you need to derive an expression involving exactly that. Try factorising T^(k+1) differently.
     
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