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Math Help - Fibonacci sequence proof

  1. #1
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    Fibonacci sequence proof

    Not sure if I'll be able to get a response in time but I'm gonna see what I can get. I have an exam in 3 hours. Even if it's not in time, I'd still like to see how to do this.

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  2. #2
    Senior Member DivideBy0's Avatar
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    Hey I reckon I can prove this, but as a forewarning, I'm definitely no expert in sequences or Fibonacci.
    I'll derive everything from F_(k+1) = F_(k) + F_(k-1).

    F_(k+1) = F_k + F_(k-1)
    (F_(k+1))^2 = (F_k)^2 + 2*F_k*F(k-1) + (F_(k-1))^2 ...Square both sides
    (F_(k+1))^2 - (F_k)^2 = 2*F_k*F_(k-1) + (F_(k-1))^2 ...Bring (F_(k))^2 to the left
    (F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(2F_k + F_(k-1)) ...Factor out F_(k-2)
    (F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(F_k + F_(k+1)) ...Simplify Brackets

    After further simplifying brackets:
    (F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(F_(k+2))

    Ta da!
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  3. #3
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    Thank you very much. I'm kinda embarassed that this mostly boiled down to simple algebra.
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  4. #4
    MHF Contributor

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    Here is short proof.
    Attached Thumbnails Attached Thumbnails Fibonacci sequence proof-fib112.gif  
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