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Sequence with nested square roots

Source: Cono Sur Olympiad 2017, problem 6

August 21, 2017
cono surnumber theory

Problem Statement

The infinite sequence a1,a2,a3,…a_1,a_2,a_3,\ldots of positive integers is defined as follows: a1=1a_1=1, and for each n≥2n \ge 2, ana_n is the smallest positive integer, distinct from a1,a2,…,an−1a_1,a_2, \ldots , a_{n-1} such that: an+an−1+…+a2+a1\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}} is an integer. Prove that all positive integers appear on the sequence a1,a2,a3,…a_1,a_2,a_3,\ldots
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