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2012 ToT Fall Senior A p1 a_k = a_{k+T}

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March 22, 2020
Sequencealgebra

Problem Statement

Given an infinite sequence of numbers a1,a2,a3,...a_1, a_2, a_3,... . For each positive integer kk there exists a positive integer t=t(k)t = t(k) such that ak=ak+t=ak+2t=...a_k = a_{k+t} = a_{k+2t} =.... Is this sequence necessarily periodic? That is, does a positive integer TT exist such that ak=ak+Ta_k = a_{k+T} for each positive integer k?
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