SOLUTION: Can you please assist me with the following problem? CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he n

Question 83790: Can you please assist me with the following problem?
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?
b) How much money expressed in dollars would the traveling salesman receive in total if the checkerboard only had 32 squares?
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?
Thank you for your help...
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If you look at the relationship between the number of pennies on the squares you'll see that the number is growing exponentially. Going from 1 to 2, the number is doubled. Continuing from 2 to 4, again the number is doubled. So to get to any term you must double the previous value. So this problem involves powers and exponents. If you wanted to get to the 10th term, you would start at the first term and double each term to get to the tenth term. To get to the 10th term, you must multiply 2 by itself 10 times. So to get to the 64th term, you must multiply 2 by itself 64 times, see the pattern? To get the pattern down officially, the sequence is where n is any term you pick. We start off at n=0 to get the first term of 1 and we move from there. Hope a little background on this helps, so here we go.

a)To find out how many pennies are on the 32th square, simply evaluate 2^31 (we go to the n-1 term since we started off at n=0). Well it comes out to 2,147,483,648 pennies.
If you want to verify, you can double 1 to 2, 2 to 4, etc until you get there.
Divide this figure by 100 to get the value in dollars.

So he would pay $21,474,836.48 alone for the 32nd square

b)To find the sum of any geometric series, i.e. how to find 1+2+4+8+...2,147,483,648, you would use the sum of a geometric series formula. If I have a series of n terms the sum is
don't worry about a, we will ignore it, a=1 right now
and r is the factor to go from term to term, in this case r=2 (we're doubling everything).
So

so this means that there are a total of 4,294,967,295 pennies if 32 squares were filled.
Divide this figure by 100 to get the value in dollars.

So if there were only 32 squares, then he would pay $42,949,672.95

c)To find how many pennies would fill the entire board, use the same formula but with 64 squares

So there are 18,446,744,073,709,551,616 pennies needed to fill the entire board. Divide this figure by 100 to get the value in dollars.

So he would have to pay $184,467,440,737,095,516.16 to the salesman.
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