Games
Problems
Go Pro!

The Chess Problem - Sequences and Series

Lesson Plans > Mathematics > Algebra > Functions > Sequences and Series > Geometric Series
 

The Chess Problem - Sequences and Series

When 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.
Lesson by Mr. Twitchell

Blogs on This Site

Reviews and book lists - books we love!
The site administrator fields questions from visitors.
Like us on Facebook to get updates about new resources
Home
Pro Membership
About
Privacy