This paper is concerned with stability of linear discrete time-delay systems. Firstly, a new finite-sum inequality is proposed, which is the further promotion for Abel lemma-based finite-sum inequality. Then, by using the inequality and constructing appropriate Lyapunov-Krasovskii functionals(LKFs), a new delay-dependent stability criteria is obtained in terms of linear matrix inequalities(LMIs). Numerical examples are given to demonstrate that the proposed method can provide a larger admissible maximum upper bound than those using the Abel lemma-based finite-sum inequality approach, and it involves less decision variables than the free-weighting matrix method.