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Bundeswettbewerb Mathematik 1994 Problem 2.2

Source: Bundeswettbewerb Mathematik 1994 Round 2

October 7, 2022
Sequencenumber theoryDivisibility

Problem Statement

Let kk be an integer and define a sequence a0,a1,a2,…a_0 , a_1 ,a_2 ,\ldots by a0=0,    a1=k    and    an+2=k2an+1−an  for  n≥0. a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0. Prove that an+1an+1a_{n+1} a_n +1 divides an+12+an2a_{n+1}^{2} +a_{n}^{2} for all nn.
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