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Part of 2017 China National Olympiad

Problems(1)

China Mathematical Olympiad 2017 Q1

Source: China Changsha ，Dec 23， 2016

11/23/2016
The sequences {un}\{u_{n}\} and {vn}\{v_{n}\} are defined by u0=u1=1u_{0} =u_{1} =1 ,un=2un−1−3un−2u_{n}=2u_{n-1}-3u_{n-2} (n≥2)(n\geq2) , v0=a,v1=b,v2=cv_{0} =a, v_{1} =b , v_{2}=c ,vn=vn−1−3vn−2+27vn−3v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3} (n≥3)(n\geq3). There exists a positive integer NN such that when n>Nn> N, we have un∣vnu_{n}\mid v_{n} . Prove that 3a=2b+c3a=2b+c.
algebranumber theorySequence