Question 969206: Suppose you work for a company that ays $0.01 the first day, $0.02 the second day, $0.04 the third day, and so on. If the daily wage keeps doubling, what would your total income be for working 30 days?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
If you double the amount every day, you will have  cents the first day,  cents the second day plus the 1 cent you got the first day, and so on. So you need the sum of the amounts that you got for each of the thirty days. So if you count the first day as day zero, and the last day as day 29, you are going to get \times 0.01) dollars which we can write as
\cdot\sum_{k=0}^{29}\,2^k)
Now you can just go at this straight up, calculating the value of each of the 30 terms and then adding them together, but there is a considerably easier way.
It turns out that
Or, in the case of this particular problem
\cdot\sum_{k=0}^{29}\,2^k\ =\ (0.01)\left(2^{30}-1\right))
You can do your own calculator work from here. But I'll make you a deal. You pay me $0.01 the first day, $0.02 cents the next, and double the amount every day for thirty days and I'll give you back a fixed $150,000.00 every day for the same thirty days. All amounts in US Dollars. The only stipulation is that we settle up at the end of the month. Let me know. (Hint: Do NOT take this deal. You would end up about six and a quarter million dollars in the hole at the end of the month)
John
My calculator said it, I believe it, that settles it
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