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Unbounded sum of sequence

Source: 2020 China North Mathematical Olympiad Advanced Level P1

August 4, 2020
Sequencesunboundedrecursionalgebra

Problem Statement

The function f(x)=x2+sin⁡xf(x)=x^2+ \sin x and the sequence of positive numbers {an}\{ a_n \} satisfy a1=1a_1=1, f(an)=an−1f(a_n)=a_{n-1}, where n≥2n \geq 2. Prove that there exists a positive integer nn such that a1+a2+⋯+an>2020a_1+a_2+ \dots + a_n > 2020.
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