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Sequnce of positive integers

Source: IZHO 2017 day 2 p4

January 15, 2017
algebranumber theoryInteger sequenceSequence

Problem Statement

Let (an)(a_n) be sequnce of positive integers such that first kk members a1,a2,...,aka_1,a_2,...,a_k are distinct positive integers, and for each n>kn>k, number ana_n is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers a1,a2,...,an−1a_1,a_2,...,a_{n-1}. Prove that an=2an−1a_n=2a_{n-1} for all sufficently large nn.
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