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a_{n+1}=a_n+ \frac{n^2}{a_n}, b_n =a_n-n

Source: China Northern MO 2022 p3 CNMO

April 6, 2024
algebrainequalitiesrecurrence relation

Problem Statement

Let {an}\{a_n\} be a sequence of positive terms such that an+1=an+n2ana_{n+1}=a_n+ \frac{n^2}{a_n} . Let bn=an−nb_n =a_n-n . (1) Are there infinitely many nn such that bn≥0b_n \ge 0 ? (2) Prove that there is a positive number MM such that ∑n=3∞bnn+1<M\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M.
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