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Math Help - Fibonacci Number

  1. #1
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    Fibonacci Number

    Suppose f n is the nth Fibonacci number. Prove that
    f1^2*f2^2 +....Fn^2 = fn*fn+1 where n is a positve integer.

    Any ideas????
    Last edited by dh214; November 5th 2008 at 02:04 PM. Reason: symbols
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  2. #2
    Super Member PaulRS's Avatar
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    Let's think geometrically.

    By definition f_{n+1}=f_n+f_{n-1}

    Let us draw a square of side-length f_1=1 first, then put another one but with side-length f_2=1 beisde the other one. So that they share a side. Now we can put a square of side f_3=2 right next to the other ones so that a border is shared and so on...

    For example see here

    Note now that, the area of the big rectangle is equal to the sum of the areas of the small squares inside. So, if the sides of the rectangle are f_n and f_{n+1}, it follows that f_{n+1}\cdot{f_n}=f_n^2+f_{n-1}^2+...+f_1^2

    You may also try using induction
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