PROBLEM 269

We build a sequence of fractions as follows:  The numerator and denominator of the first fraction are each 1.  The denominator of a new fraction is the sum of the numerator and denominator of the previous fraction while the numerator of a new fraction is the sum of the new denominator and previous denominator. In symbols, if we let n_k and d_k denote the numerator and denominator of the k'th fraction, then  n₁ = 1 = d₁, and d_k+1 = n_k + d_k and n_k+1 = d_k+1 + d_k.  Does the sequence of fractions {n₁/d₁,n₂/d₂,n₃/d₃,...} = {1/1,3/2,7/5,17/12,41/29...} converge?  If so, to what does it converge? 

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ã2007 Alberto L. Delgado