We propose a family of Hardy-type tests for an arbitrary n-partite system,which can detect different degrees of non-locality ranging from standard to genuine multipartite non-locality.For any non-signaling m-local hidden variable model,the corresponding tests fail,whereas a pass of this type of test indicates that this state is m non-local.We show that any entangled generalized GHZ state exhibits Hardy’s non-locality for each rank of multipartite non-locality.Furthermore,for the detection of m non-localities,a family of Bell-type inequalities based on our test is constructed.Numerical results show that it is more efficient than the inequalities proposed in [Phys.Rev.A 94 022110 (2016)].