On the Solutions of the Lucas Sequence Equation ± 1/Vn(P2, Q2) = Σk=1∞ Uk-1(P1,Q1)/xk

Suppose that {U-n(P, Q)} and {V-n(P, Q)} are respectively the Lucas sequences of the first and second kinds with P not equal 0, Q not equal 0 and gcd(P, Q) = 1. In this paper, we introduce an approach for studying the solutions (x, n) of the diophantine equation +/- 1/Vn(()P(2), Q(2)) = Sigma(infinity)(k=1) Uk-1(P-1,Q(1))/x(k) in the cases of (P-1, Q(1)).= (P-2, Q(2)) and (P-1, Q(1)) = (P-2, Q(2)). Moreover, we apply the procedure of this approach with which -3 <= P-1, P-2 <= 3, -2 <= Q(1) <= 2 and -1 <= Q(2) <= 1. Our approach is mainly based on transferring this equation into either an elliptic curve equation that can be solved easily using the Magma software, or a quadratic equation that can be solved using the quadratic formula.