Many authors have studied the problem of finding sequences of rational points on elliptic curves such that either the abscissae or the ordinates of these points are in arithmetic progression. In this paper we obtain upper bounds for the lengths of sequences of rational points on curves of the type y² = x³ + k, k ∈ Q \ {0}, such that the ordinates of the points are in arithmetic progression, and also when both the abscissae and the ordinates of the points are separately the terms of two arithmetic progressions.