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Math Help - An interesting fraction

  1. #1
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    An interesting fraction


    \displaystyle\frac{1}{998,001} \;=\;0.\overline{000\,001\,002\,003\,004\,005\, \hdots\,996\,997\,999}\, \hdots

    The decimal representation contains all the 3-digit numbers except 998
    . . and the 2997-digit cycle repeats forever.

    This is just one of a family of such fractions.
    Can you determine the underlying characteristic?
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  2. #2
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    Crna Gora
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    Re: An interesting fraction

    f(n)=\frac{10^n}{\left(10^n-1\right)^2}

    where n is a number of digits of integer (fractional) part of denominator .
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  3. #3
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    Re: An interesting fraction

    More interesting fractions..

    \frac{100}{9899}

    \frac{10100}{970299}

    \frac{10000}{970299}
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  4. #4
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    Re: An interesting fraction

    Hello, a tutor!

    Very nice fractions!



    \frac{100}{9899} \;=\;0.01\,01\,02\:03\;05\,08\,13\,\hdots
    . . . . . . .Fibonacci sequence


    \frac{10100}{970299} \;=\;0.01\,04\:09\,16\,25\,\hdots
    . . . . . . . . Squares


    \frac{10000}{970299} \;=\;0.01\,03\;06\,10\,15\,21\,\hdots
    . . . . . . . . Triangular numbers



    One more . . .

    \frac{1}{89} \;=\;0.0\,1\,1\,2\,3\:5\,9\,5\,5\,\hdots
    . . . . . Fibonacci sequence

    Code:
           1
          --  =  0.01
          89       001
                   0002
                   00003
                   000005
                   0000008
                   00000013
                   000000021
                       .
                       .
                       .


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