probability

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

Show that Pn satisfies the recurrence relation
Pn​=(2p−1)Pn−1​+(1−p)

Solve this for Pn​ explicitly in terms of p and n and explain why your answer makes sense for the special cases when (i)p0​ (ii)p1/2​ and (iii)p1​

I am unsure how to approach this given the information and how it is related to recurrence relations

 
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