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Arithmetic

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Course: Michigan Algebra II KHauck
Book: Arithmetic
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Date: Tuesday, November 26, 2024, 12:38 AM

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Arithmetic

Table of contents

Arithmetic

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

S5 = a1 + a2 + a3 + a4 + a5

In general, the sum of the first n terms of an arithmetic sequence is represented by:

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

Example 1

Find the sum of the first five terms (S5) of the arithmetic sequence 1, 4, 7, 10, 13 …

Step 1. Determine the number of terms to add.

S5 = a1 + a2 + a3 + a4 + a5

Step 2. Add the terms.

S5 = 1 + 4 + 7 + 10 + 13
S5 = 35

Example 2

Find S7 for the arithmetic sequence 1, 4, 7, 10, 13 …

Step 1. Determine the number of terms to add.

S7 = a1 + a2 + a3 + a4 + a5 + a6 +a7



Step 2. Continue the pattern to find the needed terms.

a 6 = 16 and a7 = 19

Step 3. Add the terms.

S7 = 1 + 4 + 7 + 10 + 13 + 16 + 19
S7 = 70
 

Sum of Finite Series

When adding a small number of terms, the process used in the previous examples is effective. When the application requires the addition of a large number of terms, this process becomes too cumbersome and a formula will accelerate the process.

Consider the sequence from the previos example.

1, 4, 7, 10, 13, 16

In this sequence, add terms 1 and 6, terms 2 and 5, and terms 3 and 4.

SumFinite1

SumFinite2

SumFinite3

Notice each pair has the same sum and there are SumFinite4 of these sums.
 

Series Formula

In general, any arithmetic sequence can be paired to find the same sum as the first and last terms and there will be SeriesFormula1 pairs. Therefore, the following formula can be used to determine the arithmetic series with any number of terms.

SeriesFormula2

SeriesFormula3 or SeriesFormula4


Where a 1 is the first term, a n is the last term, and n is the number of terms.

 

Example 1

Find the sum of the first 50 terms of an arithmetic sequence whose first term is -5 and whose 50th term is -95.

Step 1. Determine a1 and an.

SeriesFormulaEx1-1 and SeriesFormulaEx1-2

Step 2. Determine the number of pairs in the sequence.

SeriesFormulaEx1-3

Step 3. To find the series, substitute and solve.

SeriesFormulaEx1-4
SeriesFormulaEx1-5
 

Example 2

The sum of the first 50 terms of an arithmetic sequence is 4,800. If the 50th term is 194, find the first term and the common difference.

Step 1. Determine a1 and an.
SeriesFormulaEx2-1 and SeriesFormulaEx2-2
Step 2. Determine the number of pairs that will make this sum.
SeriesFormulaEx2-3
Step 3. Substitute into the formula and solve.
SeriesFormulaEx2-4
SeriesFormulaEx2-5
SeriesFormulaEx2-6
The first term, a1, is -2.
 
Step 4. Determine the formula for the sequence.
SeriesFormulaEx2-7
SeriesFormulaEx2-8

Step 5. Solve for d.
SeriesFormulaEx2-9
SeriesFormulaEx2-10

The common difference is 4.

Application

An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on. The auditorium has 30 rows of seats. How many seats are in the auditorium?

Step 1. Write an arithmetic sequence to represent the situation.

20, 24, 28, 32 …

Step 2. Determine the general formula for the sequence.

Application1-1
Application1-2
Application1-3

Step 3. Determine a1 and an. For this situation, a1 = first row and an = last row.
Application1-4 and Application1-5

Step 4. Determine the number of pairs in the sequence. In this situation there are 30 rows, so n = 30.
Application1-6
Step 5. To find the series, substitute and solve.
Application1-7
Application1-8
The auditorium has 2340 seats.

Video Lesson

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

Calculate Arithmetic Series

Guided Practice

To solidify your understanding of arithmetic 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 Arithmetic Series Worksheet

Arithmetic Series Worksheet

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

To learn more about Arithmetic Series watch the video below:

 

Answer Key

Sum of Arithmetic Series Worksheet Answer Key

Arithmetic Series Worksheet Answer Key

*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 an Arithmetic Series." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=7227/7227.xml (accessed 7/14/2010).

Holt, Rinehart & Winston, "Sequence and Series." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch12_03_homeworkhelp.html (accessed 7/14/2010).

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