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Each term of the sequence is an integer

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September 9, 2010
number theorySequencerecurrence relationIMO Shortlistalgebra

Problem Statement

Let aa be a positive integer and let {an}\{a_n\} be defined by a0=0a_0 = 0 and an+1=(an+1)a+(a+1)an+2a(a+1)an(an+1)(n=1,2,… ).a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ). Show that for each positive integer nn, ana_n is a positive integer.
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