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China South East Mathematical Olympiad 2021 Grade10 P1

Source:

July 28, 2021
Sequencealgebra

Problem Statement

A sequence {an}\{a_n\} is defined recursively by a1=12,a_1=\frac{1}{2}, and for n≥2,n\ge 2, 0<an≤an−10<a_n\leq a_{n-1} and an2(an−1+1)+an−12(an+1)−2anan−1(anan−1+an+1)=0.a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0. (1)(1) Determine the general formula of the sequence {an};\{a_n\}; (2)(2) Let Sn=a1+⋯+an.S_n=a_1+\cdots+a_n. Prove that for n≥1,n\ge 1, ln⁡(n2+1)<Sn<ln⁡(n+1).\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).
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