The force acting on a particle moving along X-axis is F = - k(x - u0 t) where k is a positive constant. An observer moving at a constant velocity u0 along the X-axis looks at the particle. What kind of motion does he find for the particle?

As the person is moving with the constant velocity, he will observe the same force. There is no relative motion between the force and the person. Hence, he will not observe the particle in motion.