An interesting fraction

**Soroban** · April 24th 2012, 04:25 PM

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

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?

**princeps** · April 24th 2012, 08:43 PM

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

where $n$ is a number of digits of integer (fractional) part of denominator .

**a tutor** · April 24th 2012, 11:44 PM

More interesting fractions..

$\frac{100}{9899}$

$\frac{10100}{970299}$

$\frac{10000}{970299}$

**Soroban** · April 25th 2012, 08:52 AM

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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