As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. Suppose that the weather is dry on January 1st and let Pn be the probability that the weather is dry n days later, so that January 1st is day n 0, January 2nd is day n 1, and so on. a.) (10 points) Show that Pn satisfies the recurrence relation Pn 2p 1Pn1 1 p for n 1, 2, 3, 4,.... b.) (10 points) Solve this for Pn explicitly in terms of p and n and explain why your answer makes sense for the special cases when (i) p 0 (ii) p 1/2 and (iii) p 1. Note in doing this, you may assume that a general solution for the equation Sn aSn1 fn for when a is a non-zero constant and fn is a known function of n is Sn an1S1 k2 n ankfk where S1 is the value of Sn when n 1 and k0 m rk rm1 1/r 1, when r 1 m 1