Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.5 AIME Prove it Belarus

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n = a_{n-1} + a_{n+1}$
a) for all $n \ge 2$ ;
b) for all $n \ge 3$ .

Solution

a) there are no such sequences;

b) there exists a unique sequence: $a_1 = a_2 = 1$ , $a_i = 2$ for all $i \ge 3$ .

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