In this article, we concentrate on common fixed points of a pair of generalized non-expansive mappings, viz., generalized α-Reich-Suzuki non-expansive mappings. In this sequel, we introduce the three step Abbas-Nazir iterative algorithm for a pair of mappings. Then we obtain some results related to weak and strong convergence of sequences, satisfying this iterative algorithm, to obtain the common fixed points of two generalized α-Reich-Suzuki non-expansive mappings. Finally, we compare the convergence rate of our iteration technique to that of some well-known iteration techniques by some constructive numerical examples.