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Math Help - Fibonacci members and powers of the golden ratio

  1. #1
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    Fibonacci members and powers of the golden ratio

    Hello. I searched for a topic for the same question, but my queries were fruitless, although they returned some similar questions. If i have missed it, please lock this one, and redirect me to the proper thread.
    It's not a homework assignment, but a personal inquiry due to some home studies I'm performing. I have a little background in maths, but haven't done anything of the sorts in quite a while ( > 4 years), and was looking for some confirmation on my work. Anyhow, here's the problem.
    Prove that ith Fibonacci number satisfies the equality
    <br />
F_n = \frac{{\phi}^n - {\hat{\phi}}^n}{\sqrt{5}}<br />
    Where  \phi is the golden ratio and  \hat{\phi} is its conjugate.

    What I've done:
    Proof by induction.
    Base cases:
    <br />
F_0 = {\phi}^0 - {\hat{\phi}}^0<br />
    <br />
\\F_0 = 0<br />

    <br />
F_1 = {\phi}^1 - {\hat{\phi}}^1<br />
    <br />
\\F_1 = 1<br />

    Inductive hypothesis:
    Suppose that it holds for [LaTeX ERROR: Convert failed]
    The same for \hat{\phi}.

    From that we trivially arrive at
    <br />
\frac{{\phi}^{n+1} - {\hat{\phi}}^{n+1}}{\sqrt5} = \frac{{\phi}^n + {\phi}^{n-1}}{\sqrt5} - \frac{{\hat{\phi}}^{n} + {\hat{\phi}}^{n-1}}{\sqrt5}<br />
    Which is what we were trying to prove. Can you please confirm my reasoning? I'm not sure if I'm allowed such an assumption in my inductive hypothesis, and also my reasoning looks a little circular. If there's a gap, can you please point it out to me, as well as any hints at getting the proper solution?
    Last edited by ossinator; May 22nd 2010 at 09:33 AM.
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  2. #2
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    Quote Originally Posted by ossinator View Post
    Hello. I searched for a topic for the same question, but my queries were fruitless, although they returned some similar questions. If i have missed it, please lock this one, and redirect me to the proper thread.
    It's not a homework assignment, but a personal inquiry due to some home studies I'm performing. I have a little background in maths, but haven't done anything of the sorts in quite a while ( > 4 years), and was looking for some confirmation on my work. Anyhow, here's the problem.
    Prove that ith Fibonacci number satisfies the equality
    <br />
F_n = \frac{{\phi}^n - {\hat{\phi}}^n}{\sqrt{5}}<br />
    Where  \phi is the golden ratio and  \hat{\phi} is its conjugate.

    What I've done:
    Proof by induction.
    Base cases:
    <br />
F_0 = {\phi}^0 - {\hat{\phi}}^0<br />
    <br />
\\F_0 = 0<br />

    <br />
F_1 = {\phi}^1 - {\hat{\phi}}^1<br />
    <br />
\\F_1 = 1<br />

    Inductive hypothesis:
    Suppose that it holds for F_n and F_{n-1}
    We also know, that \phi and \hat{\phi} are the roots of the equation  x^2 = x + 1
    Therfore, if we multiply both sides of the equation by  {\phi} ^ {n-1}, we get  {\phi}^{n+1} = {\phi}^{n} + {\phi}^{n-1}
    The same for \hat{\phi}.

    From that we trivially arrive at
    <br />
\frac{{\phi}^{n+1} - {\hat{\phi}}^{n+1}}{\sqrt5} = \frac{{\phi}^n + {\phi}^{n-1}}{\sqrt5} - \frac{{\hat{\phi}}^{n} + {\hat{\phi}}^{n-1}}{\sqrt5}<br />
    Which is what we were trying to prove. Can you please confirm my reasoning? I'm not sure if I'm allowed such an assumption in my inductive hypothesis, and also my reasoning looks a little circular. If there's a gap, can you please point it out to me, as well as any hints at getting the proper solution?
    That is a very nice proof.
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  3. #3
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    Does that mean that it works?
    Also, there's a mistake in my base cases, a \sqrt{5} is missing.
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  4. #4
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    Quote Originally Posted by ossinator View Post
    Does that mean that it works?
    Also, there's a mistake in my base cases, a \sqrt{5} is missing.
    Yes, you should have F_0 = ({\phi}^0 - {\hat{\phi}}^0)/\sqrt5 and F_1 = ({\phi}^1 - {\hat{\phi}}^1)/\sqrt5. But apart from that the proof is completely correct.
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