Recursive method to solve the problem of “Gambling with God”

Suppose Alice gambles with God who is the dealer. There are nn total rounds in the game and God can choose any mm rounds to win and the other n−mn−m rounds to lose. At first Alice has holdings aa. In each round, Alice can increase her holdings by qq times the amount she wagers if she wins. So what strategy should Alice take to ensure the maximum total holdings in the end? And how much is the total final holdings? It is called the “Gambling with God” problem. In this paper, a recursive method is proposed to solve the problem, which shows the extensive application of recursive methods.