Consider a loop with a length of 1, then randomly select a continuous part in this loop with a length of 0.5 and cover it. Repeat this process for k times until the loop is completely covered. Find the mathematical expectation of k.

My personal thoughts are as follows.

I have personally considered the question for quite some time. I tried to disagree the question considering the possibility that the cycle is completely covered with 1,2,3 times of coverage, thinking of adding them to obtain the expectation of k later. When k = 1 and k = 2, obviously the probability is 0, when k = 3 the possibility can be shown by an image. Take the starting point of the first covered line segment as the reference point and, respectively, take the second and third covered line segments as x and y in a coordinate plane. We can trace the image as shown below.

As we create all possibilities, we can have this image. The shaded area is expressed when x lands at these specific points, where y should be to meet the requirements. When calculating the area of the shadow, we can conclude that when k = 3, the probability that the loop is completely covered is 0.25

But when I started with the condition of k = 4, it implies a larger plot dimension, which is much more difficult, and I became unable to think clearly about that problem. Can anyone provide a new perspective on this problem? Or is there an elegant solution?