In computer games and also in simulations of card-playing scenarios, we sometimes need to use a computer to simulate the way that person would shuffle a deck of cards. Given two decks of n cards each, the riffle shuffle algorithm involves repeatedly choosing one of the two decks at random and then removing the bottom card from that deck and placing that card on the top of an output deck. This card-choosing step is repeated until all the original cards are placed in the output deck. Define a recursive-riffle algorithm, which cuts a deck of n cards into two decks of n/2 each, where n is a power of 2, and then calls the recursive-riffle algorithm on each half. When these recursive calls return, the recursive-riffle algorithm then performs a riffle shuffle of the two decks to produce the shuffled result. Show that every card in the original deck has an equal probability of being the top card after a recursive-riffle is performed on the deck, and analyze the running time of the recursive-riffle algorithm using a recurrence equation.