Let R be a ring and n be a positive integer. In this paper, further results on the n-strong Drazin inverse are obtained in a ring. We prove that a ∈ R is n-strongly Drazin invertible if and only if a-an+1 is nilpotent. In terms of this characterization, the extensions of Cline's formula and Jacobson's lemma for this inverse are proved. Moreover, the n-strong Drazin invertibility for the sums of two elements is considered. We prove that a,b ∈ R are n-strongly Drazin invertible if and only if a + b is n-strongly Drazin invertible, under the condition ab = 0. As applications for the additive results, we obtain some equivalent conditions of the n-strong Drazin invertibility of matrices over a ring.