Based on geometrical-optics approximation (GOA) of forward scattering by a gradient refractive index (GRIN) sphere, a further extension of the method to the on-axis Gaussian beam scattering of a transparent gradient refractive index spherical particle is developed. The scattering angle of Gaussian beam through the GRIN sphere and the phase shift are deduced and the calculation procedure of the method is given. On the basis of the above work, the scattering intensity distribution is calculated by the extension of the GOA and the result is compared with those calculated by the generalized Lorenz-Mie theory (GLMT). When the beam waist is larger than the particle radius, the discrepancy exists because of the surface wave influence. Compared with our results, the extension of GOA is valid on condition that the beam-waist radius is smaller than the radius of the particle. The comparison between calculation speed of GOA method and the one of the (GLMT) is given. The result indicates that the calculation speed of GOA method is faster.