This problem is similar to the Noah’s Descendents problem, except you need the sum of all of the 40 doors, not just the cost of the final one. So, starting with a table to look for a pattern:
door number cost (cents) total (cents)
1 10 10
2 20 30
3 40 70
4 80 150
If you look at the cost column, it is just repeatedly multiplying by 2 (which is really the same as using exponents): 80 = 10 x 2 x 2 x 2 =
10 x 2^3 (4th term is the 3rd power). The 5th term would be 10 x 2^4, and the 40th term would be 10 x 2^39 cents.
The total column is just 2 x cost column minus 10 cents, so the total would be (2 x 10 x 2^39 - 10) cents = (10 x 2^40 - 10)/100 dollars, which is $109,951,162,800. If you calculated the cost of the 40th door instead of the total, you found that it cost $54,975,581,390.
That would be a very expensive house today!
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