**Introduction**

To determine the value of n for which the average velocity of a particle, moving in a straight line and subjected to a force F = -kvn, is one-third of its initial velocity, we need to analyze the motion of the particle and use the concept of average velocity.

**Analysis**

Let's consider the motion of the particle under the influence of the given force. According to Newton's second law, the force acting on the particle is equal to the rate of change of its momentum:

F = dp/dt

Since momentum is defined as the product of mass and velocity (p = mv), we can rewrite the equation as:

F = d(mv)/dt

Using the product rule of differentiation, we can expand the equation as:

F = m(dv/dt) + v(dm/dt)

Now, we know that the mass of the particle remains constant, so dm/dt is zero. Therefore, the equation simplifies to:

F = m(dv/dt)

**Determining the Value of n**

Given that F = -kvn, we can substitute it into the equation:

-kvn = m(dv/dt)

Rearranging the equation, we have:

dv/vn = -k/m dt

To find the average velocity over a period of time, we integrate both sides of the equation:

∫dv/vn = ∫-k/m dt

Using the power rule of integration, we get:

(1/n) ∫v^(-n) dv = (-k/m) ∫dt

Integrating the left side with respect to v and the right side with respect to t, we obtain:

(1/n) * (v^(-n+1))/(1-n) = -(k/m) * t + c

where c is the constant of integration.

**Finding the Average Velocity**

To find the average velocity of the particle, we need to determine the time it takes for the particle to come to rest (v = 0). At that point, the constant c becomes zero as well.

Setting v = 0 and solving for t, we have:

(1/n) * (0^(-n+1))/(1-n) = -(k/m) * t

Since anything raised to the power of zero is 1, the equation simplifies to:

0 = -(k/m) * t

Since k and m are positive constants, we can conclude that t = 0.

**Conclusion**

From our analysis, we can see that for the average value of velocity to be one-third of the initial velocity, the value of n must be 3.