The correct answer is option 3: the force acting on a particle executing simple harmonic motion is directly proportional to the displacement and is directed towards the mean position.
In simple harmonic motion, the restoring force that acts on the particle is proportional to its displacement from the mean position. This means that the farther the particle is from the mean position, the stronger the force pulling it back towards the mean position.
Option 1 states that the force is directly proportional to the displacement but is directed away from the mean position. This is incorrect because the force in simple harmonic motion always tries to restore the particle back to its equilibrium position.
Option 2 states that the force is inversely proportional to the displacement and is directed towards the mean position. This is also incorrect, as the force is actually directly proportional to the displacement, not inversely proportional.
Option 4 is incorrect because it states that the force is inversely proportional to the displacement and is directed away from the mean position. Again, this goes against the nature of simple harmonic motion, where the force always acts towards the mean position.
Therefore, the correct answer is option 3: the force acting on a particle executing simple harmonic motion is directly proportional to the displacement and is directed towards the mean position.