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essentially different Fibonacci sequences

Source: Netherlands - Dutch NMO 1971 p2

January 28, 2023
fibonacci numberSequencecombinatoricsnumber theory

Problem Statement

A sequence of real numbers is called a Fibonacci sequence if tn+2=tn+1+tnt_{n+2} = t_{n+1} + t_n for n=1,2,3,..n= 1,2,3,. . . Two Fibonacci sequences are said to be essentially different if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences 1,2,3,5,8,...1,2,3,5,8,... and 1,3,4,7,11,...1,3,4,7,11,... are essentially different, but the sequences 1,2,3,5,8,...1,2,3,5,8,... and 2,4,6,10,16,...2,4,6,10,16,... are not.
(a) Prove that there exist real numbers pp and qq such that the sequences 1,p,p2,p3,...1,p,p^2,p^3,... and 1,q,q2,q3,...1,q,q^2,q^3,... are essentially different Fibonacci sequences.
(b) Let a1,a2,a3,...a_1,a_2,a_3,... and b1,b2,b3,...b_1,b_2,b_3,... be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence t1,t2,t3,...t_1,t_2,t_3,..., there exists exactly one number α\alpha and exactly one number β\beta, such that: tn=αan+βbnt_n = \alpha a_n + \beta b_n for n=1,2,3,...n = 1,2,3,...
(c) t1,t2,t3,...t_1,t_2,t_3,..., is the Fibonacci sequence with t1=1t_1 = 1 and t2=2t_2= 2. Express tnt_n in terms of nn.
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