The purpose of this work is to generalize the concepts of k-solvability and k-nilpotency, initially defined for n-Lie algebras, to n-Hom-Lie algebras and to study their properties. We define k-derived series, k-central descending series and study their properties, we show that k-solvability is a radical property and we apply all of the above to the case of (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras.