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Romania District MO 2022 Grade 11 P3

Source: Romania District MO 2022 Grade 11

March 27, 2022
Sequencelimitcollege contestsromania

Problem Statement

Let (xn)n≥1(x_n)_{n\geq 1} be the sequence defined recursively as such: x1=1, xn+1=x1n+1+x2n+2+⋯+xn2n ∀n≥1.x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.Consider the sequence (yn)n≥1(y_n)_{n\geq 1} such that yn=(x12+x22+⋯xn2)/ny_n=(x_1^2+x_2^2+\cdots x_n^2)/n for all n≥1.n\geq 1. Prove that
[*]xn+12<yn/2x_{n+1}^2<y_n/2 and yn+1<(2n+1)/(2n+2)⋅yny_{n+1}<(2n+1)/(2n+2)\cdot y_n for all n≥1;n\geq 1; [*]lim⁡n→∞xn=0.\lim_{n\to\infty}x_n=0.
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