[Math Lair] Exponential Growth

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You may have run into a problem such as the following, or perhaps you tried this trick on your parents when you were young (if you're still young, here's your chance):

You have to wash the dishes each day next month. Which would you rather have: $10 per day, or 1¢ the first day, and double the previous day's amount on every other day (so 2¢ the second day, 4¢ the third day, and so on)?

The second option doesn't really look like much, but if you calculate it out, your payment for the 30th day will be $5,368,709.12!

You may also recall the legend of the invention of chess: According to legend, when the inventor of the game showed it to the King of Persia, the King was impressed and asked him what he would like as a reward for the invention. The inventor just asked for one grain of wheat for the first square on the chessboard, two grains of wheat for the second, four for the fourth, and so on, doubling the amount for each square. The king thought that to be a small reward, but calculation would discover that 18,446,744,073,709,551,615 would be required!