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Thread: Recursive sequence (Fibonacci Sequence) & Limits

  1. #1
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    Recursive sequence (Fibonacci Sequence) & Limits

    Hey, I would like some help with this question:

    A recursive sequence is a sequence where the $\displaystyle n$th term, $\displaystyle t_{n}$, is defined in terms of preceding terms, $\displaystyle t_{n-1}, t_{n-2}$, etc

    one of the most famous recursive sequences is the Fibonacci sequence, created by Leonardo Pisano (1170-1250). The terms of this sequence are define as follows $\displaystyle f_{1} = 1, f_{2} = 1, f_{n} = f_{n-1} + f_{n-2}$ where $\displaystyle n \geq 3$

    It asks me to graph some things, did those, but then it asks me to write an expression, using a limit, to represent the value of the ratios of consecutive terms of the Fibonacci sequence. How would I go about doing this?

    Thanks in advance.
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  2. #2
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    Hi

    $\displaystyle f_{n} = f_{n-1} + f_{n-2}$

    $\displaystyle \frac{f_{n}}{f_{n-1}} = 1 + \frac{f_{n-2}}{f_{n-1}}$

    $\displaystyle \frac{f_{n}}{f_{n-1}} = 1 + \frac{1}{\frac{f_{n-1}}{f_{n-2}}} = 1 + \frac{1}{1 + \frac{1}{\frac{f_{n-2}}{f_{n-3}}}}$
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  3. #3
    MHF Contributor chisigma's Avatar
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    Setting $\displaystyle \rho_{n} = \frac{f_{n}}{f_{n-1}}$ the sequence of the $\displaystyle \rho_{n}$ is generated recursively by the relation...

    $\displaystyle \rho_{n}= 1 + \frac{1}{\rho_{n-1}}$ (1)

    ... that is equivalent to the difference equation...

    $\displaystyle \Delta_{n} = \rho_{n} - \rho_{n-1}= 1 - \rho_{n-1} + \frac{1}{\rho_{n-1}}$ (2)

    The (2) indicates that the limit when n tends to infinity of the $\displaystyle \rho_{n}$ is given by the positive solution of the equation...

    $\displaystyle \rho^{2} - \rho - 1 = 0$ (3)

    ... so that is...

    $\displaystyle \lim_{n \rightarrow \infty} \rho_{n} = \frac{1 + \sqrt {5}}{2} = \varphi = 1.61803398874989\dots$ (4)

    The constant $\displaystyle \varphi$ is known as 'golden ratio'...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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