Suppose you are given a job.... problem

**the** · January 17th 2010, 10:32 AM

Suppose you're given a job that lasts for one month (31 days). You get one cent on the first day, two cents on the second, 4 on the third, etc.

What's the total amount you would make?

I know the formula to calculate the amount has something to do with 2^(n-1) for the amount received on the last day... and I know that (2^n)-1 has something to do with summarizing the total amount, but I'm not sure where to go.

I'm not looking for an outright answer, but just something to understand why those formulas are necessary and how to head off in the right direction. Thanks!

**Dinkydoe** · January 17th 2010, 10:48 AM

Well:

day one you get: $2^{0}= 1$
day 2 you get : $2^{1} = 2$
day 3 you get: $2^{2} = 4$

So the last day you get $2\cdot$ (the amount of 30-th day) = $2^{31-1} = 2^{30}$

And the total amount you get is the sum of the amounts you earn everyday:

total amount = $\sum_{n=1}^{31}2^{n-1} = 1+2+4+\cdots + 2^{30}$

**Dinkydoe** · January 17th 2010, 12:37 PM

I'd like to add: we can recognise this serie as a sum of the form: $\sum_{k=1}^{n}ar^{k-1} = \sum_{k=0}^{n-1}ar^k= a\frac{1-r^n}{1-r}$ where a= 1, r = 2, n= 31.

More about such series on wikipedia: Geometric series - Wikipedia, the free encyclopedia

Thus the total amount = $2^{31}-1$

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