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Checkerboard Problem: Sums of Geometric Sequences

Get your checkerboard and place one grain of wheat on the first square. Then place two grains of wheat on the next square. Then place four grains on the third square. Continue this until all 64 squares are covered with grains of wheat." As he had just harvested his wheat, Mr. Crane did not consider this much of an award, but he soon realized he made a miscalculation on the amount of wheat involved.

a) How much wheat would Mr. Crane have to put on the 24th square?
b) How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?
c) Calculate the amount of wheat necessary to fill the whole checkerboard (64 squares). How much wheat would the farmer need to give the salesman? Please provide the answer in either scientific notation, or calculate and show all 20 digits.

Solution Preview

a) How much wheat would Mr. Crane have to put on the 24nd square?
Answer: 8388608 grains
Show work in this space
an = 1*2^n-1
a24 = 2^24-1 = ...

Solution Summary

Sums of geometric sequences are investigated. The four grains on the third square are determined.

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