In this paper, we present numerical and experimental results on Localized Surface Plasmon Resonance (LSPR) refractive index (RI) sensitivity, Figure of Merit (FoM), and penetration depth (d_p) dependence on spherical gold nanoparticles (AuNPs) size, and the effects of AuNP dimer interparticle distance (d_s) studied numerically. These parameters were calculated and observed for d = 20, 40, 60, 80, and 100 nm diameter spherical AuNPs. Our investigation shows d = 60 nm AuNPs give the best FoM. The AuNP dimer interparticle distance can significantly influence the RI sensitivity. Therefore, the effect of distances between pairs of d = 20 nm and 60 nm AuNPs is shown. We discuss the importance of penetration depth information for AuNPs functionalized with aptamers for biosensing in the context of aptamer size.