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Thread: Continued Fraction Questions

  1. #1
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    Continued Fraction Questions

    1) Let $\displaystyle f_{k}$ denote the kth Fibonacci number. Find the simple continued fraction, terminating with the partial quotient of 1, of $\displaystyle f_{k+1}/f_{k}$, where k is a positive integer.

    2) Show that if $\displaystyle a_{0} > 0$, then
    $\displaystyle p_{k}/p_{k-1} = [a_{k}; a_{k-1},.....,a_{1}, a_{0}]$
    and
    $\displaystyle q_{k}/q_{k-1} = [a_{k}; a_{k-1},.....,a_{2}, a_{1}]$,
    where $\displaystyle C_{k-1} = p_{k-1}/q_{k-1}$ and $\displaystyle C_{k} = p_{k}/q_{k}, k \geq 1$, are successive convergents of the continued fraction $\displaystyle [a_{0};a_{1},...,a_{n}]$
    (Given a hint for this one: Use the relation $\displaystyle p_{k} = a_{k}p_{k-1}+p_{k-2}$ to show that $\displaystyle p_{k}/p_{k-1} = a_{k} + 1/(P_{k-1}/p_{k-2}$).)
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  2. #2
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    The Fibonnacci numbers are

    $\displaystyle \displaystyle 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$.


    $\displaystyle \displaystyle \frac{f_2}{f_1} = \frac{1}{1} = 1$

    $\displaystyle \displaystyle \frac{f_3}{f_2} = \frac{2}{1} = 1 + \frac{1}{1}$

    $\displaystyle \displaystyle \frac{f_4}{f_3} = \frac{3}{2} = 1 + \frac{1}{1 + 1}$

    $\displaystyle \displaystyle \frac{f_5}{f_4} = \frac{5}{3} = 1 + \frac{1}{1 + \frac{1}{1 + 1}}$

    $\displaystyle \displaystyle \frac{f_6}{f_5} = \frac{8}{5} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1+1}}}$.


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