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a_n is integer

Source: Iranian National Olympiad (3rd Round) 2002

October 1, 2006
number theory proposednumber theory

Problem Statement

ana_{n} is a sequence that a1=1,a2=2,a3=3a_{1}=1,a_{2}=2,a_{3}=3, and an+1=an−an−1+an2an−2a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}} Prove that for each natural nn, ana_{n} is integer.
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