SOLUTION: thanks for any help 4) 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

Algebra ->  Algebra  -> Sequences-and-series -> SOLUTION: thanks for any help 4) 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      Log On

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Question 53425: thanks for any help
4) 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 Brown came out and gratefully thanked the traveling salesman for saving his daughter’s life. Mr. Brown 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 penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies.” As he’d been saving pennies for over 25 years, Mr. Brown did not consider this much of an award, but soon realized he made a miscalculation on the amount of money involved.
a) How much money expressed in dollars would Mr. Brown have to put on the 32nd square?
Answer:
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b) How much money expressed in dollars would the traveling salesman receive in total if the checkerboard only had 32 squares?
Answer:
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c) Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How money expressed in dollars would the farmer need to give the salesman?
Answer:
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Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
See if this makes sense to you! If you double the number of pennies on each succeeding square, starting with one penny, the progression would look like this:
Square...1...2...3...4...5...6... ...n
Pennies..1...2...4...8...16..32.. ..2%5E%28n-1%29
a) So then, on the 32nd square, n = 32 and the number of pennies would be: 2%5E%2832-1%29+=+2%5E31 = 2,147,483,648 pennies = $21,474,836.48
Answer a) is: $21,474,836.48
b) The total number of pennies required to fill a 32-square checker board would be:
1+2+4+8+16+...+2%5E%2831%29 = 2%5E32+-+1+=+4294967296+-+1 = 4,294,967,295 pennies = $42,949,672.95
c) The total number of pennies required to fill a 64-square checker board would be:
2%5E64+-+1 = 18,446,744,073,709,551,615 pennies = $184,467,440,737,095,516.15