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Math Help - Prove Conjecture

  1. #1
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    Prove Conjecture

    Prove conjecture for the sum of the squares of the first n Fibonacci numbers

    (F1) ^2 + (F2) ^ 2 .... + (Fn) ^2

    Thanks for the help!!
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  2. #2
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    What is the conjecture?
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  3. #3
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    well I'm really trying to prove

    f1^2 + F2^2 +.... Fn^2 = (Fn) (Fn+1) by assuming n = K + 1

    Thanks for Speedy response
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  4. #4
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    This is certainly an inductive proof. But it is not difficult.

    We have:

    1^2 = (1)(1)

    1^2 + 1^2 = (1)(2)

    1^2 + 1^2 + 2^2 = (2)(3)

    So it looks like the first few cases are true.

    Suppose \sum _{k=1} ^n (F_k)^2 = F_nF_{n+1} is true.

    Then \sum _{k=1} ^{n+1} (F_k)^2 = F_nF_{n+1} + (F_{n+1})^2 = (F_n + F_{n+1})(F_{n+1}) = F_{n+2}F_{n+1} = F_{n+1}F_{n+2}.

    And the induction is complete.
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  5. #5
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    ahh thanks for the help
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