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Geometric

Site: Clare-Gladwin RESD
Course: Michigan Algebra II KHauck
Book: Geometric
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Date: Tuesday, November 26, 2024, 12:49 AM

Description

Geometric

Table of contents

Geometric

Adding the terms of the sequence is required to solve many applications of sequences. The sum of the terms of a sequence is called a series. A geometric series is the sum of the terms of a geometric sequence. For example, the sum of the first five terms of a geometric sequence is represented by:

S5 = a1 + a2 + a3 + a4 + a5

In general, the sum of the first n terms of a geometric sequence is:

Sn = a1 + a2 + a3 + a4 + ... + an

For a geometric sequence with first term a1, a common ratio of r, and n terms the series is given by the formula:

Geometric1-1
 

Example 1

Find the sum of the first 7 terms of the geometric sequence -3, 6, -12, 24, -48 …

Step 1. Determine a1, r and n.
GeometricEx1-1 , GeometricEx1-2 , and GeometricEx1-3

Step 2 . To find the series, substitute and solve.
GeometricEx1-4

GeometricEx1-5

GeometricEx1-6

GeometricEx1-7
 

Example 2

Find the sum of the first 20 terms of the geometric sequence:

2, 6, 18, 54 …

Step 1. Determine a1, r and n.
GeometricEx2-1 , GeometricEx2-2 and GeometricEx2-3

Step 2. To find the series, substitute and solve.
GeometricEx2-4

GeometricEx2-5

GeometricEx2-6

GeometricEx2-7
 

Application

Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second day, 4 cents on the third day, etc. If the daily wage continues to double, what is your total income for working 31 days?

Step 1. Write a geometric sequence to represent the situation.

1, 2, 4, 8, 16 …

Step 2. Determine a1, r and n.
GeoApp1-1, GeoApp1-2 and GeoApp1-3

Step 3. Determine the general formula for the sequence .

GeoApp1-4

GeoApp1-5

Step 4. To find the series, substitute and solve.
GeoApp1-6

GeoApp1-7

GeoAPp1-8

GeoApp1-9

The total pay for 31 days is 2,147,483,647¢ or $21,474,836.47.

Video Lesson

To learn how to calculate geometric series, select the following link:

Calculate Geometric Series

Guided Practice

To solidify your understanding of geometric series, visit the following link to Holt, Rinehart, and Winston Homework Help Online. It provides examples, video tutorials, and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Practice

Sum of Geometric Series Worksheet

Geometric Series Worksheet

*Note: If Google Docs displays, “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

To learn more about Geometric Series watch the video below.

Answer Key

Answer Key for Sum of Geometric Series Worksheet

Answer Key for Geometric Series Worksheet

*Note: If Google Docs displays, “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Sources

Sources used in this book:

Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project

Florida Virtual School, http://www.flvs.net/ (accessed 2/25/2010).

Holt, Rinehart & Winston, "Finding the Sum of a Geometric Series." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=7232/7232.xml (accessed 7/14/2010).

Holt, Rinehart & Winston, "Sequences and Series." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch12_04_homeworkhelp.html (accessed 7/14/2010).

Stapel, Elizabeth. "Arithmetic and Geometric Sequences." http://www.purplemath.com/modules/series3.htm (accessed 2/25/2010).