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Funky Fibonacci Fractions (Posted on 2007-02-22) Difficulty: 4 of 5
100/89= 1.1235..., which includes the first five Fibonacci numbers.

10000/9899=1.010203050813213455... includes the first ten Fibonacci numbers.

1000000/998999=1.001002003005... produces the first 15 Fibonacci numbers.

If you add two zeros in the numerator and two nines (one at the beginning, one at the end) in the denominator, does this Fibonacci production go on?

See The Solution Submitted by Old Original Oskar!    
Rating: 4.6667 (3 votes)

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Solution Proof Comment 5 of 5 |
The fractions are 10^2k/(10^2K-10^K-1), so writing z=10^K, we have z^2/(z^2-z-1). Finally, if we let x=1/z, we get f(x)=1/(1-x-x^2), which is a generating function for the Fibonacci numbers: f(x)=ΣFnx^n, and since x=10^-K, the result follows.
  Posted by Federico Kereki on 2007-02-23 08:34:44
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