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Geometric Series and Sums : Checkerboard Story Problem - 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 ....

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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 checker-board 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.
How much money would Mr. Brown have to put on the 32nd square?
How did you solve it for the answer?

How much would the traveling salesman receive if the checkerboard only had 32 squares?
Again show me how you solved it.

Calculate the amount of money necessary to fill the whole checker board (64 squares). How much money would the farmer need to give the salesman?

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Solution Preview

This is a Geometric Series problem.

In general, on the n'th square, ...

Solution Summary

A geometric series problem is solved.

$2.19
See Also This Related BrainMass Solution

Arithmetic Sequences

1) Use the arithmetic sequence of numbers 1, 3, 5, 7, 9,...to find the following:
a) What is d, the difference between any 2 terms?
Answer:
Show work in this space.

b) Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer:
Show work in this space.

c) Using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms?
Answer:
Show work in this space

d) Using the formula for the sum of an arithmetic series, what is the sum of the first 30 terms?
Answer:
Show work in this space

e) What observation can you make about these sums of this series (HINT: It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)? Express your observations as a general formula in "n."
Answer:

2) Use the geometric sequence of numbers 1, 2, 4, 8,...to find the following:
a) What is r, the ratio between 2 consecutive terms?
Answer:
Show work in this space.

b) Using the formula for the nth term of a geometric sequence, what is the 24th term?
Answer:
Show work in this space.

c) Using the formula for the sum of a geometric series, what is the sum of the first 10 terms?
Answer:
Show work in this space

3) Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,...to find the following:
a) What is r, the ratio between 2 consecutive terms?
Answer:
Show work in this space.

b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Please round your answer to 4 decimals.
Answer:
Show work in this space.

c) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Please round your answer to 4 decimals.
Answer:
Show work in this space.

d) What observation can make about these sums? In particular, what number does it appear that the sum will always be smaller than?
Answer:

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:
Show work in this space

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

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

Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a function?
Include the following in your answer:
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic sequence?
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric sequence?
? Give at least two examples of a sequences or series. One example should be arithmetic, and the second should be geometric.

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