A runner starts running on a circular path of radius 'r'

I believe the last sentence should say: What is the ratio of the time taken for the nth round to that for the previous round? Without stating time, the question can't really be answered.

Given that information, though:

Time = Distance / Velocity

In the first interval: Velocity = (∏r meters/min) => (∏r/2 meters in 30 seconds) => Quarter-Lap in 30 seconds: T = 30 seconds, D = Quarter-Lap, Velocity = ∏r

Since the velocity is being halved and the time is being doubled, the distance is clearly constant: D = V * T = (V_old * 0.5) * (T_old * 2) = D_old

So at each velocity and time shift, we are covering a quarter-round of distance. We need to compare the times to complete 4-interval rounds.

Since T = D/V, and D is constant, T will vary inversely with V.

Consider how the velocity for a quarter round in one cycle will change with respect to a quarter round in the next cycle:

Since the velocity is halved each time, it will be sixteenth-ed after each series of 4 quarter-rounds. (1/2 * 1/2 * 1/2 * 1/2 = 1/16).

So since each round's velocity is 1/16 of the previous round's velocity, the time is multiplied by 16.

So, -C-.