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VMO 2022 problem 1 day 1

Source: Vietnam Mathematical Olympiad 2022 problem 1 day 1

March 4, 2022
vmocalculusSequencealgebra

Problem Statement

Let aa be a non-negative real number and a sequence (un)(u_n) defined as: u1=6,un+1=2n+an+n+anun+4,∀n≥1u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1 a) With a=0a=0, prove that there exist a finite limit of (un)(u_n) and find that limit b) With a≥0a \ge 0, prove that there exist a finite limit of (un)(u_n)
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