The problem on linear scattering of a train of femtosecond laser pulses by a weakly absorbing spherical particle is solved based on the analytical solution of the Maxwell equations derived through representation of light fields as a series expansion in terms of natural electromagnetic modes of a dielectric sphere. The evolution of the optical field in the particle as it is exposed to a single pulse or a train of laser pulses is analyzed comparatively. It is found that when the particle is exposed to a series of femtosecond laser pulses, the evolution of the particle's internal field and its intensity varies depending on the gap between the pulses. This effect is shown to be connected with the excitation of natural electromagnetic modes (whispering gallery modes) in the particle with the resonance frequencies falling within the spectrum of the initial laser pulse. There exists an optimal gap between the pulses, at which the intensity of the internal optical field additionally increases in the zone of its maximum. The value of this gap is inversely proportional to the relative frequency mismatch between the natural modes excited and the central frequency of the incident radiation.