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On Fibonacci Sequence

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September 15, 2010
algebraFibonacciFibonacci sequenceSequenceIMO Shortlistnumber theory

Problem Statement

Let {fn}\{fn\} be the Fibonacci sequence {1,1,2,3,5,… .}.\{1, 1, 2, 3, 5, \dots.\}.
(a) Find all pairs (a,b)(a, b) of real numbers such that for each nn, afn+bfn+1af_n +bf_{n+1} is a member of the sequence.
(b) Find all pairs (u,v)(u, v) of positive real numbers such that for each nn, ufn2+vfn+12uf_n^2 +vf_{n+1}^2 is a member of the sequence.
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