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Math Help - Fibonacci Numbers

  1. #1
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    Fibonacci Numbers

    Q: Show that the Fibonacci numbers F(1),F(2),F(3),... ,where F(1)= F(2)=1 and F(k)= F(k-2) + F(k-1) for k>2 satisfy the following equality for all n (greater or equal to) 1.

    (F(1))^2 + (F(2))^2 + (F(3))^2+.....+ (F(n))^2 = F(n) * F(n+1)


    I did the basis step:
    1^2 + 1^2 = 1

    but how do I do the inductive step for this question?



    Thanks,
    Creative
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  2. #2
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    Quote Originally Posted by Creative View Post
    Q: Show that the Fibonacci numbers F(1),F(2),F(3),... ,where F(1)= F(2)=1 and F(k)= F(k-2) + F(k-1) for k>2 satisfy the following equality for all n (greater or equal to) 1.
    (F(1))^2 + (F(2))^2 + (F(3))^2+.....+ (F(n))^2 = F(n) * F(n+1)

    I did the basis step:
    1^2 + 1^2 = 1
    You did not do the first step.
    1^2 + 1^2 \ne 1
    Is it true that 1^2 + 1^2 = F(2)F(2+1)?
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  3. #3
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    Oops basis step is 1^2(F1) = 1^2(F2) = 1?

    1^2 +1^2 = F(2)*F(2+1)
    is not true?
    1^2*1^2? = 1
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  4. #4
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    Quote Originally Posted by Creative View Post
    Oops basis step is 1^2(F1) = 1^2(F2) = 1?
    1^2 +1^2 = F(2)*F(2+1) is not true?
    Yes it is true.
    1^2 + 1^2=2
    F(2)=1\;\&\;F(2+1)=F(3)=2.
    F(2)F(3)=2
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  5. #5
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    Thanks, so basis step would be proving F(1)
    F(1)*(F(1+1)) = F(1)*F(2) = (1^2)*(1^2) = 1
    so true for basis step.
    Then how would I do inductive?
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  6. #6
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    I wonder if you understand what the problem really says?
    You want to prove \sum\limits_{k = 1}^n {\left[ {f(k)} \right]^2 }  = f(n)f(n + 1).
    We have more or less done the first step.
    Now assume that \sum\limits_{k = 1}^J {\left[ {f(k)} \right]^2 }  = f(J)f(J + 1) is true.
    On that basis prove it is true for J+1.
    I am not going to do it for you. For it is your problem.
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