3.9.7 Families in a certain society choose the number of children that they will have according to the following rule: If the first child is a girl, they have exactly one more child. If the first child is a boy, they continue to have children until the first girl and then cease childbearing. Let  be the number of male children in a particular family. What is the generating function of ? Determine the mean of  directly and by differentiating the generating function.