The correct answer is A. a simple harmonic motion.
A simple harmonic motion is a periodic motion that is characterized by a restoring force that is proportional to the displacement from the equilibrium position. In the case of the particle attached to the elastic string, the restoring force is the force of gravity, which is proportional to the weight of the particle. The equilibrium position is the position where the particle is not moving, and the displacement is the distance from the equilibrium position.
When the particle is pulled down vertically through a distance $x$, it is displaced from the equilibrium position. The restoring force then acts to return the particle to the equilibrium position. The particle will oscillate back and forth between the equilibrium position and the displaced position. The frequency of the oscillation is determined by the mass of the particle and the stiffness of the string.
The motion of the particle is not a rectilinear motion with constant speed. A rectilinear motion is a motion in a straight line. The speed of the particle is not constant, as it changes as the particle moves up and down. The motion of the particle is also not a damped oscillatory motion. A damped oscillatory motion is an oscillatory motion that is gradually decreasing in amplitude. The motion of the particle is not damped, as it does not gradually decrease in amplitude.