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Sequence of positive integers, are they all equal 2?

Source: Kazakhstan National Olympiad 2024 (9 grade -- P6), (10-11 grade -- P5)

March 21, 2024
algebra

Problem Statement

An integer m≥3m\ge 3 and an infinite sequence of positive integers (an)n≥1(a_n)_{n\ge 1} satisfies the equation an+2=2an+1m−1+anm−1m−an+1.a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. for all n≥1n\ge 1. Prove that a1<2ma_1 < 2^m.
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