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 grain of wheat on the first square. Then place two
grains of wheat on the next square. Then place four grains of wheat on the
third square. Continue this until all 64 squares are covered with grains of
wheat.” 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 whaet involved.
a) How much wheat would Mr. Brown have to put on the 24nd square?
Answer: 
Show work in this space 

1 + 2 + 4 + 8 + ···

This is a geometric sequence with r = 2, because each
term is 2 times as much as the preceding term.  The
formula for the nth term is
     
an = a1rn-1
       
For a1 = 1, r = 2, and n = 24, substituting gives

a24 = (1)(2)24-1  

a24 = 223 

a24 = 8,388,608 grains of wheat on the 24th
square alone.


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

1 + 2 + 4 + 8 + ···

The formula for the sum of the first
n terms is

      a1(rn-1)
Sn = ----------
        r-1

For a1 = 1, r = 2, and n = 24, substituting gives

       (1)(224-1)
S24 = ------------
          2-1

       224-1
S24 = -------
         1

S24 = 224-1

S24 = 16777216-1

S24 = 1677215 grains of wheat

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 answer in either scientific notation, or calculate and show all 20 
digits
Answer: 
Show work in this space 

      a1(rn-1)
Sn = ----------
        r-1

For a1 = 1, r = 2, and n = 64, substituting gives

       (1)(264-1)
S64 = ------------
          2-1

       264-1
S64 = -------
         1

S64 = 264-1

S64 = 18446744073709551616 - 1

S64 = 18446744073709551615

or

18,446,744,073,709,551,615 grains of wheat

eighteen quintillion, 446 quadrillion, 
744 trillion, 73 billion, 709 million,
551 thousand 615 grains of wheat,
which is more wheat than exists on the 
entire planet.  In fact more wheat
than has ever been grown on this 
planet since this planet has been in
existence!

Edwin