Question 30035: A man tels another, "Get your checkboard and place one grain of wheat on the first square. Then place two on the next, then place four on the third square. Continue this until all 64 squares are covered with grains of wheat" As he had just harvested his wheat, the man did not consider this much, but he soon realized he made a miscalculation on the amount of wheat involved.
a) How much wheat would the man need to have put on the 24nd square? (show work)
b) How much total grains would the traveling salesman receive if the checkboard only had 24 squares?
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! The number of grains on each checkerboard is,
1 2 4 8 16 32 64 ...
or, in exponential notation,
2^0 2^1 2^2 2^3 2^4 2^5 2^6 ...
So the number of grains on the nth square is,
Un = 2^(n-1)
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a) on the 24th square, U24 = 2^(24-1) = 2^23
Ans:2^23
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b) this is a geometric progression.
The sum of the terms in a GP is given by
Sn = u0(r^n - 1)/(r - 1)
where u0 is the first term in the series and r is the common ratio.
here u0 = 1 and r = 2 and n is the number of terms in the series.
so,
Sn = 1(2^n - 1)/(2 - 1)
Sn = 2^n - 1
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when n = 24 squares,
S24 = 2^24 - 1
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