Question 73675This question is from textbook college algebra
: My 15 year old daughter is having some difficulty with this as it will be on her mid-term! Can you please help her? Thank you:)
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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This question is from textbook college algebra
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! This problem involves a geometric progression. A geometric progression is a sequence
in which each term after the first is obtained by multiplying the same fixed number,
called the common ratio, by the preceding term.
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I'll use the following definitions:
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a is the first term
r is the common ratio
n is the number of terms
L is the last term
S is the sum of all the terms up to the last term
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and the equations we will use are:
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For the last term use the equation: 
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For the sum up to the last term the equation is 
a) How much money expressed in dollars would Mr. Brown have to put on the 32nd square?
Answer:
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the squares contain in sequence are 1, 2, 4, 8, 16, .... all the way up to the 32nd term.
From this we can see that:
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a (the first term) is 1
r (the common ratio) is 2 ... double each preceding term to get the next term
n (the number of terms) is 32
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To find the 32nd number in this series use the equation:
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Substitute 1 for a, 2 for r, and 32 for n to get:
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Now it's a calculator problem ... use a scientific calculator to raise 2 to the 31st power.
You may have a key that looks like  . If you do, enter 2 on your calculator and
press the key. Then enter 31 and press the the = key. The answer you should
get is 2147483648 and that is the number of pennies on the 32nd square. To convert this
to dollars just move the decimal two places to the left to get $21,474,836.48 and that
is the money that is on the 32nd square only.
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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:
Show work in this space
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Use the sum equation with the values noted previously.
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substitute 1 for a, 2 for r and 32 for n
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.
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4294967295 is the total number of pennies on the board for the first 32 squares. Convert this
to the number of dollars on the board by dividing by 100 to get:
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$42,949,672.95
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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:
Show work in this space
.
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But 2^64 = 18446744073709551616 so that -(1-2^64 )= -1 + 18446744073709551616 = 18446744073709551615
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But this is the total number of pennies on the 64 square board. Convert this to dollars by
dividing by 100 to get $184,467,440,737,095,516.15
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Hope this helps your daughter figure this stuff out.
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