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Math Help - Fraction problem

  1. #1
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    Fraction problem



    1) Under what conditions will F1 and F2 be equal?

    2) If F1 and F2 are equal, what is their value?
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  2. #2
    MHF Contributor
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    Talking

    I'll rename, for simplicity: the first continued fraction I'll call "x", and the other I'll call "y".

    Note that x = A + B/(A + (B/(A + B/...)))) = A + B/[A + B/(A + (B/(A + (B/...)))) = A + B/x. Similarly, y = B + A/y.

    For these to be equal, we must have A + B/x = B + A/y. What can you get from this?
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  3. #3
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    Hello, dxdy!

    I have a good start on it . . . I'll let you finish it.


    Let A and B be distinct positive real numbers and consider the continued fractions:

    . . {\color{blue}[1]}\;\;F_1 \;=\; A + \frac{B}{A + \dfrac{B}{A + \hdots}}\qquad\qquad{\color{blue}[2]}\;\; F_2 \;=\;B + \frac{A}{B + \dfrac{A}{B + \hdots}}

    1) Under what conditions will F_1 = F_2 ?

    From [1]: . F_1 \;=\; A+ \frac{B}{F_1} \quad\Rightarrow\quad F_1^2 - AF_1 - B \:=\:0

    . . Quadratic Formula: . F_1 \;=\;\frac{A \pm\sqrt{A^2+4B}}{2}


    From [2]: . F_2 \;=\;B + \frac{A}{F_2} \quad\Rightarrow\quad F_2^2 - BF_2 - A \:=\:0

    . . Quadratic Formula: . F_2 \;=\;\frac{B \pm\sqrt{B^2 + 4A}}{2}


    \text{Since }F_1 = F_2\!:\;\;\frac{A \pm\sqrt{A^2+4B}}{2} \;=\;\frac{B \pm\sqrt{B^2+4A}}{2}

    . . . . . . A - B \;=\;\pm\sqrt{B^2+4A} \mp \sqrt{A^2+4B}

    . . . . . . A - B \;=\;\pm\left(\sqrt{B^2+4A} - \sqrt{A^2+4B}\right)


    Square both sides:

    . . A^2 - 2AB + B^2 \;=\;B^2 + 4A - 2\sqrt{A^2+4B}\sqrt{B^2+4A} + A^2 + 4B

    . . . . \sqrt{A^2+4B}\sqrt{B^2+4A} \;=\;2A + 2B + AB


    Square both sides:

    . . (A^2+4B)(B^2+4) \;=\;(2A+2B+ AB)^2

    . . A^2+B^2 + 4A^3 + 4B^3 + 16AB \;=\;4A^2 + 8AB + 4A^2B + 4B^2 + 4AB^2 + A^2B^2


    This simplifies to: . A^3 - A^2B - AB^2 + B^3 \;=\;A^2 - 2AB + B^2

    . . Factor: . A^2(A-B) - B^2(A-B) \;=\;(A-B)^2 \quad\Rightarrow\quad (A-B)(A^2-B^2) \;=\;(A-B)^2

    . . Factor: . (A-B)(A-B)(A+B) \:=\:(A-B)^2 \quad\Rightarrow\quad (A-B)^2(A+B) \:=\:(A-B)^2


    Since A \neq B, we can divide by (A-B)^2\!:\;\;\boxed{A + B \;=\;1}


    Can you finish it?

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  4. #4
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    F1 = 1/2
    f2 = 1/2
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