By using group representation theory, th quasi-adiabatic approximation solution of the schrodinger equation of a quantum system with slowly-changing Hamiltonian are presented in this paper. We not only obtained the Berry phase factor and strictly proved the quantum adiabatic theorem as the zeroth-order approximation, but also studied the universal Berry phase factor and its geometrical interpretation when the adiabatic condition is violated. It is pointed out that this universal Berry phase factor has observable effects.