A million dollars or a penny doubled every day?

Question

You are offered a choice. Option A: one million dollars, paid today. Option B: one cent today, two cents tomorrow, four cents the day after, and so on, doubling every day for thirty days, keeping everything you are given. Which option is worth more, and by how much? How much does the last day alone pay?

Accepted Answer

Take the penny. It is worth about 10.7 million dollars, more than ten times the million. The thirtieth day. Day 1 pays 1 cent, day 2 pays 2, day k pays 2 to the (k minus 1). Day 30 pays 2 to the 29 cents = 536,870,912 cents, about 5.4 million dollars on that single day. The total. 1 + 2 + 4 + ... + 2 to the 29 = 2 to the 30 minus 1 = 1,073,741,823 cents, which is 10,737,418 dollars and 23 cents. Half of the total arrives on the last day, a quarter on the day before, and the first three weeks together contribute less than 2 per cent. The rough version. 2 to the 10 is about a thousand, so 2 to …

A million dollars or a penny doubled every day?

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