Probability

A dice is thrown repeatedly until a '6' appears.
Let an = Event that a '6' appears for the 1st time on the nth throw.
Let Yr = Event that a '1' appears r times before a '6' appears for the 1st time.
(a) Show that the probability of Yr = Σ [(probability of an + 1 ) x (probability of Yr | an + 1) ], where n from r to infinity. [ Yr | an + 1 means Yr given an + 1 has happened.]
(b) Using the result of (a), show that the probability of Yr = (1/2)^(r + 1).