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IMO LongList 1988 sequence

Source: IMO LongList 1988, France 1, Problem 9 of ILL

October 22, 2005
limitquadraticsalgebra unsolvedalgebra

Problem Statement

If a0a_0 is a positive real number, consider the sequence {an}\{a_n\} defined by: an+1=an2−1n+1,n≥0. a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. Show that there exist a real number a>0a > 0 such that: i.) for all a0≥a,a_0 \geq a, the sequence {an}→∞,\{a_n\} \rightarrow \infty, ii.) for all a0<a,a_0 < a, the sequence {an}→0.\{a_n\} \rightarrow 0.
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