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Math Help - continued fraction approximations

  1. #1
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    continued fraction approximations

    Find the first 3 continued fraction approximations (as fractions not continued fractions) to:
    1.(1+(squareroot 5))/2 (golden ratio)
    2.(e-1)/2
    Can anybody help with how to go about this as im not sure and the notes i have are not clear.
    Thanks
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  2. #2
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    Hello, studentsteve1202!

    Find the first 3 continued fraction approximations to:

    1)\;\;\frac{1+\sqrt{5}}{2} . (golden mean)

    2)\;\;\frac{e^{-1}}{2}

    I know a very primitive method of doing this . . .

    \frac{1+\sqrt{5}}{2} \;=\;1.618033989\;=\;1 + 0.618033989 \;=\;1 + \frac{1}{\frac{1}{0.618033989}} \;=\;1 + \frac{1}{1.618033988}<br />

    . . . . . =\;1 + \frac{1}{1 + 0.618033988} \;=\; 1 + \frac{1}{1 + \frac{1}{1.61803988}} \;=\;1 + \frac{1}{1 + \frac{1}{1 + 0.618039988}}


    We see that the Golden Mean is:

    . . \phi \;=\; 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \hdots}}}


    The first three continued fractions are:

    a_1 \;=\;1

    a_2 \;=\; 1 + \frac{1}{1} \;=\;2

    a_3 \;=\;1 + \frac{1}{1 + \frac{1}{1}} \;=\;\frac{3}{2}


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