You own casino #2. A bright young employee suggests a new game. Players will pay a fee to enter. Then they engage in a series of 50/50 coin tosses. If their first tail occurs on turn 1, they get $1. If their first tail is on turn 2, they get $2. If their first tail is on turn 3, they get $3. If it's on turn 4, they get $4. Their potential winnings at each turn are equal to the turn number. They can flip until they flip a tail, or they can declare a tail at a turn they have successfully arrived at, and take the winnings for that turn. After they flip a tail or declare a tail, they take their winnings and the game is over.
You know what your target profit margin is, but to establish a profitable fee for this game, you need to know the break-even point; the point at which the income from the fee covers, at least, the payouts to winners over time. How much would you need to charge as an entry fee, to at least, break even in this game, over time?