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Putnam 2006 A3

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December 4, 2006
Putnamvectorlinear algebramatrixmodular arithmeticcollege contests

Problem Statement

Let 1,2,3,…,2005,2006,2007,2009,2012,2016,…1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots be a sequence defined by xk=kx_{k}=k for k=1,2…,2006k=1,2\dots,2006 and xk+1=xk+xk−2005x_{k+1}=x_{k}+x_{k-2005} for k≥2006.k\ge 2006. Show that the sequence has 2005 consecutive terms each divisible by 2006.
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