Not an answer, since I haven't worked it out. But here is how I see the problem setup.
The probability that he buys at the n-th shop is
You also know that .
Hopefully, all this could be used to solve the problem.
Jack is walking around in Luna Park, and notices an alleyway called 'Infinite Ice Cream'. Jack notes that the 'Infinite Ice-Cream' appears to possess an infinitely large number of people selling ice-cream. Upon walking outside any particular shop, Jack feels a huge compulsion to purchase an ice-cream. For every shop that Jack visits, he is less likely to purchase an ice-cream then at the previous shop. After purchasing an ice-cream, Jack leaves Luna Park. What is the probability of Jack purchasing an ice-cream at the second shop he visits?
Thanks guys !