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Math Help - Continued Fraction Questions

  1. #1
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    Continued Fraction Questions

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

    2) Show that if a_{0} > 0, then
    p_{k}/p_{k-1} = [a_{k}; a_{k-1},.....,a_{1}, a_{0}]
    and
    q_{k}/q_{k-1} = [a_{k}; a_{k-1},.....,a_{2}, a_{1}],
    where C_{k-1} = p_{k-1}/q_{k-1} and C_{k} = p_{k}/q_{k}, k \geq 1, are successive convergents of the continued fraction [a_{0};a_{1},...,a_{n}]
    (Given a hint for this one: Use the relation p_{k} = a_{k}p_{k-1}+p_{k-2} to show that 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 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots.


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

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

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

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

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


    Starting to see a pattern?
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