We present an ab initio study of the excited state properties of silicon nanoparticles (NPs) with diameters of 1.2 and 1.6 nm. Quasiparticle corrections were computed within the G(0)W(0) approximation. The absorption spectra were computed by time-dependent density functional theory (TDDFT) using the adiabatic PBE approximation, and by solving the Bethe-Salpeter equation (BSE). In our calculations, we used recently developed methods that avoid the explicit inversion of the dielectric matrix and summations over empty electronic states. We found that a scissor operator reliably describes quasiparticle corrections for states in the low energy part of the spectra. Our results also showed good agreement between the positions of the absorption peaks obtained using TDDFT and the BSE in the low part of the spectra, although the peak intensities differ. We discuss the effect of the Tamm-Dancoff approximation on the optical properties of the NPs and present a quantitative analysis in terms of sum rules. In the case of the BSE we found that, even in the absence of the Tamm-Dancoff approximation, the f-sum rule is not fully satisfied due to an inconsistency between the approximations used for the BSE kernel and for the quasiparticle Hamiltonian.