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 − a n+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.