LOGO - PULSE

Take Your Medicine 1

Author: Brink Harrison
Adapted from Section 9.1 in Hughes-Hawlett, Deborah, et.al; Single Variable Calculus; John Wiley & Sons, Inc.; New York; 2002


Time: 1 to 2 days
Preparation Time: none
Materials: none

Abstract

Students learn to change from the expanded form of a finite geometric series to the closed form of a finite geometric series and calculate the sum of the series. They use the closed form to calculate the amount of a specific antibiotic remaining in a body after a given number of doses have been taken and graph the amount of the drug in the body in between doses over time. From this graph students will infer the cumulative amount of the drug in the body as the number of doses taken increases and the see the importance of taking the full regimen of the antibiotic to combat antibiotic-resistance infections.


Objectives

(Day 1) Students will able to:

• Change from the expanded form of a finite geometric series to the closed form to facilitate calculating the sum of the series.

(Day 2) Students will able to:

• Use the sum of a finite geometric series to calculate the amount of antibiotic in the body just after the nth dose is taken, given the percentage of the original given amount of the antibiotic is present at the end of a given time period.

• Use the sum of a finite geometric series to calculate the amount of antibiotic in the body just before the nth dose is taken, given the percentage of the original given amount of the antibiotic is present at the end of a given time period.

• Draw a graph that shows the quantity of the antibiotic in the body between doses as a function of time.

• Draw a graph that shows the cumulative amount of the antibiotic in the body over time.

• Explain why the graph approaches a constant amount of antibiotic in the body.

Math Standards
 Numbers and Operations
Compute fluently and make reasonable estimates
Algebra
Use mathematical models to represent and understand quantitative relationships
Problem Solving
Apply and adapt a variety of appropriate strategies to solve problems
Connections
Recognize and apply mathematics in context outside of mathematics


Teacher Background
This lesson should follow an introductory lesson where the students are asked to identify geometric series as well as determining the value of a and being able to calculate the value of r. They have not found the sum of a geometric series yet.

A geometric series is a series in which each consecutive term is a multiple of the one before. If the first term of the series is a and the constant multiplier, or common ratio of successive terms is r, then a finite geometric series with n terms has the form:

Notice that the exponent of the last term is (n-1). This is because when you include the first term a, you want to have a total of n terms,

We will use the symbol,
                                     

to represent the sum of the first n terms of a finite geometric series. The expanded form of the sum of the first n terms of a finite geometric series is written as the equation,

The purpose of the first part of the lesson is to be able to convert the expanded form of a finite geometric series to the closed form to make it easier to calculate the sum of a series with many terms. The closed form formula for the sum of the first n terms of a finite geometric series is:

where a is the first term of the series, r is the common ratio between consecutive terms, and n is the number of terms in the series.

  

 

Activity

Have the students to discuss what they know about geometric series. (See teacher background)

Tell the students that they will use the symbol,
                                           
to represent the sum of the first n terms of a finite geometric series. Have the students write the expanded form for the following series:

Ask them how they would find the sum of the following series:

If there are not many terms, all they need to do is add the terms together. Have them discuss what they would do if there were 40 terms or 100 terms or n terms. The following procedure will lead the students to the formula for the closed form of any finite geometric series.

To find the sum of any finite geometric series, we write the sum of the first n terms as:

Multiply this equation by r to get:

Now subtract the second equation from the first:

We now have the equation:

Factoring out common terms, we have the equation:

Solving, we end up with the equation:

This is called the closed form for the first n terms of a finite geometric series, and it’s much easier to use to calculate the sum than to add all of the terms together. All the students need to know are the value of a (the first term), the value of r (the common ratio between successive terms) and the value of n (the number of terms in the finite series).

Examples: For the following finite geometric series, determine the value of a, the value of r,

the value of n, and calculate the sum of the series by using the closed form.



Homework

For the following finite geometric series, determine the value of a, the value of r, the
value of n, and the value of the last term in the series. Then calculate the sum of the series by using the closed form of the series.


8) Calculate the following quotients:

The answers to these problems are the same as the answers to problems 2 – 6 above. Why do you think this happens?

9) Think of some real-life application where you would need to calculate the sum of a finite geometric series.

 

Embedded Assessment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PULSE is a project of the Community Outreach and Education Program of the Southwest Environmental Health Sciences Center and is funded by:


an NIH/NCRR award #16260-01A1
The Community Outreach and Education Program is part of the Southwest Environmental Health Sciences Center: an NIEHS Award

LOGO - SWEHSC LOGO - NIEHS Center LOGO - NIEHS

Supported by NIEHS grant # ES06694

© 1996-2007, The University of Arizona
Last update: March 7, 2007 		  Page Content: Rachel Hughes
Web Master: Travis Biazo