Lesson 6
1. Lesson 6
1.11. Connect
Module 1: Sequences and Series
Open the Lesson 6 Assignment you saved in your course folder at the beginning of this lesson. Complete the assignment.
Project Connection
In this lesson you learned how infinite geometric series can be used to calculate the value of perpetuities. In your Module 1 Project you will interview someone who is retired. You will then begin to develop your own retirement plan.
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Go to Module 1 Project: Financing Your Dreams. Complete Project Connection 6: Retirement.
Going Beyond
In mathematics, a proof is a series of statements that are logically related to one another and that, as a whole, establish the truth of a claim. The recreational mathematician and writer Martin Gardner once said the following:
There is no more effective aid in understanding certain [proofs] than a good diagram . . . . In many cases, a proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.
Source: Martin Gardner. “Mathematical Games,”
Scientific American, October 1973.
The diagram shown is an example of what Gardner was talking about. The diagram illustrates a proof of the formula for any infinite geometric series with a first term equal to 1. Note the two statements below the diagram. The first is a statement based on the diagram. The second is a natural outcome of the first statement.
Study the diagram. Describe the details of the proof.
- What is the logic behind the diagram?
- What does the first statement mean?
- How does the second statement follow from the first?
- How do the statements relate to the diagram?
Did You Know?
Recreational mathematics focus on the math underlying games and puzzles. Despite never taking any math course after high school, Martin Gardner maintained a deep interest in mathematics.
c 2.0 Mathematisches Forschungsinstitut Oberwolfach gGmbH – attribution - sharealike
Gardner was the author of the “Mathematical Games” column for the publication Scientific American for almost three decades. He wrote articles on topics such as flexagons, tangrams, cryptanalysis, and fractals. Did you know that fractals may be described as shapes whose properties can be described by geometric series?
Gardner passed away on May 22, 2010, at the age of 95.
Search the Internet for more “proofs without words.” You may find there are some very elegant ways of demonstrating a mathematical truth by using only a well-constructed diagram. Share your discoveries with classmates or your teacher.
Save your work in your course folder.