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all elements of the sequence are integers

Source: Bundeswettbewerb 2003, Second Stage, problem 2

January 17, 2004

algebra solvedalgebra

Problem Statement

The sequence

\{a_1,a_2,\ldots\}

is recursively defined by

a_1 = 1

a_2 = 1

a_3 = 2

, and

a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0.

Prove that all elements of the sequence are integers.