In the generalized gradient approximation (GGA), energies, electronic structures and magnetic properties have been investigated for a single Ni atomic chain encapsulated in an armchair single-wall (n,n) carbon nanotubes (5≤n≤9) by using the first-principles projector augmented wave (PAW) potential within the density function theory (DFT) framework. The results show that the (5,5) tube is too narrow to wrap a Ni atomic chain, but the (6,6) tube is the smallest one to wrap a Ni atomic chain, especially at its central axis due to the lowest formation energy. The analyses of the spin-polarized band structures, total density of states (DOS), partial density of states (PDOS) and the magnetic moment of Ni@(6,6) and Ni@(7,7) systems show that the 3d states of Ni atom play determinant rales in DOS at the Fermi level, and the broader carbon nanotubes restrict slightly the magnetic moment of Ni atomic chain compared with with the narrower ones.