A gambler repeatedly bets that a die will come up 6 when rolled. Each time the die comes up 6, the gambler wins $1; each time it does not, the gambler loses $1. He will quit playing either when he is ruined or when he wins $300. If is the probability that the gambler is ruined when he begins play with then for every integer k with Also and Find an explicit formula for and use it to calculate (Exercise 33 in Section 9.9 asks you to derive the recurrence relation for this sequence.)
To find an explicit formula for and use it to calculate .
The given recurrence relation is
On solving the above characteristic equation the solutions are
The general solution of the recurrence relation is of the form
Now let us apply the initial conditions
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