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a_{n+1} =a^2_n /(a^2_n - a_n + 1)

Source: Canada Repêchage 2024/4 CMOQR

March 25, 2024
algebrarecurrence relationinequalities

Problem Statement

A sequence {ai}\{a_i\} is given such that a1=13a_1 = \frac13 and for all positive integers nn an+1=an2an2−an+1.a_{n+1} =\frac{a^2_n}{a^2_n - a_n + 1}. Prove that 12−132n−1<a1+a2+...+an<12−132n,\frac12 - \frac{1}{3^{2^{n-1}}} < a_1 + a_2 +... + a_n <\frac12 - \frac{1}{3^{2^n}} , for all positive integers nn.
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