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Chinese Girls Mathematical Olympiad 2016, Problem 5

Source: China Beijing ,13 Aug 2016

August 14, 2016
Sequenceinequalities

Problem Statement

Define a sequence {an}\{a_n\} byS1=1, Sn+1=(2+Sn)24+Sn(n=1, 2, 3, ⋯ ).S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots). Where SnS_n the sum of first nn terms of sequence {an}\{a_n\}. For any positive integer nn ,prove thatan≥49n+7.a_{n}\ge \frac{4}{\sqrt{9n+7}}.
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