ON THE REDUCTION OF A LINEAR RECURRENCE OF ORDER r

can be reduced to a relation of form (1), between xn and xn+1. Consequently, the relation's two consecutive terms of Fibonacci, Lucas, and Pell sequences are given in [2]. S. Roy [6] finds this relation for the Fibonacci sequence using hyperbolic functions. In this paper we shall prove that if a sequence (xn)n>1 satisfies a linear recurrence of order r > 2, then there exists a polynomial relation between any r consecutive terms. This shows that the linear recurrence of order r was reduced to a nonlinear recurrence of order v 1. From a practical point of view, for r > 3, expressing xn in the function of xn_l9 ..., xn_r+1 is difficult, because we must solve an algebraic equation of degree > 3 and choose the "good solution." If v = 2, we can do this in many important cases. An application of this case is a generalization of the result given in [3].