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Problem of the week #41 - January 7th, 2013

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Jameson

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Jan 26, 2012
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A very rich business woman gives you the option of taking $5,000,000 today or taking 1 penny on day 1, 2 pennies on day 2, 4 pennies on day 3.... all the way up to 30 days. Which option yields more money? Find an equation that models the total sum after n days for the second option (i.e. what is the total after 5 days? 1+2+4+8+16=31 cents)
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Jameson

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Jan 26, 2012
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Congratulations to the following members for their correct solutions:

1) dwsmith
2) veronica1999
3) Sudharaka

Solution (modified slightly from Sudharka):
The number of pennies for each day increases as a geometric progression. On day one \(2^0\) pennies are taken, on day two \(2^1\) pennies are taken and so on. Therefore on the the nth day \(2^{n-1}\) pennies are taken. So the total number of pennies taken after n days,

\[S_n=2^0+2^1+2^2+\cdots+2^{n-1}=\frac{1-2^n}{1-2}=2^n - 1\]

Therefore the sum of the money taken after 30 days,

\[S_{30}=\frac{2^{30}-1}{100}=10,737,418.23>5,000,000\] (we divide by 100 because there are 100 pennies in 1 dollar)

So better to take the second option!
 
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