Prove that all the terms of the sequence are integral
Source: CWMO 2003, Problem 5
December 27, 2008
calculusintegrationinductionfunctionquadraticsalgebra unsolvedalgebra
Problem Statement
The sequence satisfies a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots, where is a fixed positive integer. Prove that all the terms of the sequence are integral and that divides a_{2n}, n \equal{} 0, 1, 2, \ldots.