IMO 1976 LL GDR20

Let (an), n = 0, 1, . . ., be a sequence of real numbers such that

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IMO 1976 LL GDR20

Origin: GDR

Problem

Let (an), n = 0, 1, . . ., be a sequence of real numbers such that a0 = 0 and a3 n+1 = 1 2a2 n −1, n = 0, 1, . . .. Prove that there exists a positive number q, q < 1, such that for all n = 1, 2, . . ., |an+1 −an| \leqq|an −an−1|, and give one such q explicitly.