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The Chess Problem - Sequences and SeriesWhen students first start dealing with arithmetic and geometric series, it's good to give them a "startling" illustration to help them see the difference between the two types of sequences/series. In particular, it's good to help them see how quicly a geometric series can "blow up". Here's a story that helps illustrate the difference between an arithmetic series and a geometric series.
The king of Loolooland was under attack by bandits in the Looloo Forest, and was rescued by the brave Sir Lagbehind, a knight of the Rhomboid Table. The king was so grateful that he promised the knight a great reward.
"See this chessboard," the king said, pulling a chessboard out of his voluminous traveling robes, "I'm going to give you this chessboard. But before I do, I'm going to put money on each square of the board. You get to decide what I put on the squares.
"I can put a thousand dollars on the first square, two thousand on the second square, three thousand on the third square, and so on, adding a thousand for each square...
"Or I can put one penny on the first square, two pennies on the second, four on the third, and so on, doubling the amount for each square..."
The question is, which would you choose?
The instinctive reaction is: take the thousand, rather than the penny. But, of course, you can probably guess that this isn't the correct answer. The first is an arithmetic series, and the second is a geometric series. If your students have learned enough about arithmetic series and geometric series to do the calculations for themselves, you can have them figure it out themselves and report back. If not (which may be the case if you are using this as an introductory illustration), you can simply give them the results:
Starting with one thousand dollars, and adding an extra thousand each day: $2,080,000
Starting with one penny, and doubling the amount paid each day: $1.8x1017
Obviously, the second method, involving a geometric sequence, is going to break the bank!
A similar example can be used, in which a boy is hired to do a job for thirty days. He tells his boss "You can pay me twenty dollars a day, or you can pay me a penny the first day, two pennies the second day, four pennies the third day, and so on..."
Note that in this example, the first payment method involves an arithmetic sequence with difference zero. You can, of course, modify that to make it more interesting.
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fawn75 says:
There seems to be an error in this problem. The story is asking for a cumulative amount on the board. In the first situation, a cumulative amount is shown: $1,000 is added to $2,000 added to $3,000, etc. to total 2,080,000. For the second situation: If a penny is placed in box square, two pennies in the second, four in the third, and so forth... the answer given 1.8 X 10^17 is incorrect. This is only how many pennies there would be in the 64th box ALONE. However, from the story, we are being asked for the total money on the chessboard. The total number would be much greater. Instead, it should be the sum of .01 + .01x2 + .01X2^2 + .01X2^3 + ... .01x2^64.
Maybe the confusion is because the author did not mean to have a cumulative result. The reason I question this is because we are told that the first situation is arithmetic and second is geometric. If I understand correctly, arithmetic would require a linear growth. Geometric is exponential growth. However, in the first situation, according to the answer of 2,080,000, this is NOT an arithmetic sequence. If it was an arithmetic sequence, term 64th should be $64,000 only ($1,000 per box). The answer of $2,080 means the rate of change is NOT constant-- hence not arithmetic. In this case, the second situation answer would be fine in that the 64th term would be .01 X 2^64.
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