The inequality technique plays an important role in stability analysis for time-delay systems. This technical note presents a new sequence of novel summation inequalities by introducing some free matrices, which includes the newly-developed Wirtinger-based and free-matrix-based summation inequalities as special cases. Moreover, the idea can be easily extended to the multiple-summation-inequality case. Based on the proposed inequalities, relaxed stability conditions are obtained for systems with time-varying delay. Numerical examples are given to demonstrate the effectiveness of the proposed approach.