This paper is concerned with the vibrations of a mass connected to a wall or ground via a linear spring and a viscous damper arranged in series. Discrete models used to investigate the vibrations of machinery and structures often consist of a mass supported by a spring and damper in parallel or a series combination of such a system. However, models in which the spring and damper are connected in series are rarely encountered. To the authors' knowledge, there are no publications solely focused on the vibrations of a mass supported by a spring and a damper in series. In this paper, we demonstrate that the vibrational behaviour of such a system is entirely different from a system that comprises a mass with a parallel spring and damper combination. In contrast to the parallel combination, vibrations occur in this system when the damping ratio is greater than one. Another distinct result is that when the ratio of excitation frequency to the natural frequency of the system is set to zero, the frequency response function of the system with a serial combination goes to infinity. Thus, it does not correspond to the static loading case, unlike the system with a parallel combination. Furthermore, the transmissibility curves in such a system do not intersect at a certain value of the ratio of frequencies, whereas in the system with a parallel combination, these curves intersect when the ratios of frequencies are equal to 2. The system under consideration can suppress natural vibrations even for very small damping ratios, implying that this system can be used to control horizontal vibrations of a structure.