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Sequence with strange recurrency

Source: 239 Open MO, 2018, Junior League, Problem 7

April 4, 2023
algebrainequalities

Problem Statement

The sequence ana_n is defined by the following conditions: a1=1a_1=1, and for any n∈Nn\in \mathbb N, the number an+1a_{n+1} is obtained from ana_n by adding three if nn is a member of this sequence, and two if it is not. Prove that an<(1+2)na_n<(1+\sqrt 2)n for all nn.
Proposed by Mikhail Ivanov
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