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A Markov sequence is a non-zero sequence of complex numbers that satisfies a homogeneous linear difference equation with constant coefficients. The terms of such a sequence M admit of a representation of the form M(k) = cAkb, fc = 0,l,2,..., where ^4,6, c are matrices of orders n x n, n x 1, 1 x n, respectively, and n is least in the sense that b,cl are cyclic vectors for A and its transpose A1, respectively. In this article, we study the relationship between the long-term behaviour of M and the spectral properties of A. In particular, we determine the asymptotic behaviour of M(k) as k ?y co, and prescribe various conditions on M from which it can be decided that the spectral radius of A is one of its eigenvalues. For instance, we prove that this occurs when the terms of M are eventually nonnegative.