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Numerade Educator

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Problem 24 Medium Difficulty

Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer?
I. One million dollars at the end of the month.
II. One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, $ 2^{n - 1} $ cents on the nth day.

Answer

The second option is better.

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Video Transcript

all right in this problem, we want to discuss our preference between being paid $1 million at the end of a month or being paid to to the n minus one since per day. Where in is the day number? Okay, there's some things that are unclear about this problem. For example, are you working every single day in the month? Are you working 30 year, 31 or 29 days in a month, or are you just working five days a week? And so are you working more like 20 days a month? And I think we'll find that whether you're working 20 days, a month or 30 days a month or somewhere in between think you'll find that the answer is going to come out the same way? Either way you go. And the other thing is, we're assuming that what you would prefer is to make more money. So under that assumption, we're going to see which one makes more money. So let's find out what will you would get on day, um, 20. So if n equals 20 just assuming that maybe you work 20 days out of the month, how many pennies would you get that day. So this would be to raise to the 19th power, and that number is 524,288 pennies. So how many dollars is that? That would be $5242.88 on day 19. Remember, you've already made money on all the previous days. This is just day Nineteens money. Well, what if you worked? Um, 30 days. So if n equals 30 then we have two to the 29th power. Let's see what that is. Okay, that's significantly bigger. So that's 53 billion. Let me count my spots here. That's 536 million 870,912 pennies. So that's $5,368,709.12 just on day just on that one day alone, Day 30 loan. Not even counting all the money that you made on all the previous days. So I think it's pretty clear, especially because we're going to have to go with the assumption that we're talking about a full month of 30 days. It's pretty clear that getting paid in cents per day is the way to go. So the second method makes more money