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Answer by Kroki for On the asymptotic behavior of a sequence[Bulgaria MO 2024...
$x_1$ needs to be $>0$ otherwise $x_n = 0$.Using AM-GM inequality you can prove that $x_n^2 \le \sum_{k=1}^{n-1} x_k$ use induction and prove $x_n \le nx_1$ and $y_n$ is bounded.Now let $\beta :=...
View ArticleOn the asymptotic behavior of a sequence[Bulgaria MO 2024 Regional Round, 12,2]
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. We aim to prove that...
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