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Prove that [1612,2012] doesn't appear in a_n

Source: 2012 Indonesia Round 2.5 TST 3 Problem 1

May 21, 2012
inductionalgebra proposedalgebrabinary representationSequencerecurrence relation

Problem Statement

The sequence aia_i is defined as a1=2,a2=3a_1 = 2, a_2 = 3, and an+1=2an−1a_{n+1} = 2a_{n-1} or an+1=3an−2an−1a_{n+1} = 3a_n - 2a_{n-1} for all integers n≥2n \ge 2. Prove that no term in aia_i is in the range [1612,2012][1612, 2012].
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