SOLUTION: CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulle

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Question 48566: CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Crane came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Crane insisted on giving the man an award for his heroism.
So, the salesman said, “If you insist, I do not want much. 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 24nd square?
Answer:
Show work in this space.



b) How much total grain would the traveling salesman receive if the checkerboard only had 24 squares?
Answer:
Show work in this space.



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.
Answer:

Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
The number of grains of wheat that the sales is requesting can be represented as follows: 1 one the first square, 2 on the second square, 4 on the third square...and so on.
1%2B2%5E1%2B2%5E2%2B2%5E3%2B2%5E4+...+2%5E62%2B2%5E63
The number of grains on square n is: 2%5E%28%28n-1%29%29
A) So the number of grains on the 24th square would be: 2%5E%28%2824-1%29%29+=+2%5E23 = 8,388,608
B) The total number of grains on a board with only 24 squares is: 2%5E24-1 = 16,777,215
C) The total number of grains required to fill a 64-square checker board would be: 2%5E64-1 = 18,446,744,073,709,551,615