55. In the preceding problem let Mk = E[X]. Derive a recursive equation for Mk and then solve.

Hint: Start by writing Xk = Xk−1 + Ak−1,k , where Xi is the total number of successes attained up to the first time there have been i consecutive successes, and Ak−1,k is the additional number of successes after there have been k − 1 successes in a row until there have been k successes in a row.