Lesson 1

Lesson 1: Arithmetic Sequences

 

Focus

 

Are you saving money? Do you have a savings account or do you keep your coins in a safe place at home? Perhaps you want to attend college or university in the future. You may want to visit a foreign country after you graduate, or you may want to invest in a nice car. Whether you want to save for your education or finance an automobile, saving money is easy to do and can help you reach your goals.

 

There are many ways to save money. One of the easiest ways to save is to put aside a portion of your allowance or earnings from a part-time job. How much you put aside depends on how quickly you want to reach your goals. Some people like to put aside a set amount, like $50 each month. If you followed this practice of saving, can you figure out how much you will have saved at the end of two years?

 

In this lesson you will learn about arithmetic sequences. If you deposited $50 in your bank account every month, the monthly balances would form an arithmetic sequence. You will develop and learn strategies to figure out your balance at any point in the future. Doing so allows you to design a savings plan that will help you meet your financial goals.

 

Outcomes

 

At the end of this lesson you will be able to

• identify any assumptions made when defining an arithmetic sequence
  
  
• provide and describe an example of an arithmetic sequence
  
  
• determine a rule for finding the general term of an arithmetic sequence
  
  
• determine t1, d, n, or tn in a problem that involves an arithmetic sequence

Lesson Questions

 

In this lesson you will investigate the following questions:

• How can you find any term in an arithmetic sequence?
  
  
• How are patterns important in the real world?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 1 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.

 

Materials and Equipment

 

You will need a calculator.

 

Time

 

Each lesson in Mathematics 20-1 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

 

This time estimation does not include time required to complete Going Beyond activities or the Module 1 Project.

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

 

Before beginning this lesson you should be able to

• define and provide examples of natural numbers
  
  
• identify and apply patterns when given a table of values
  
  
• rearrange and evaluate formulas

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1.  
   1. What is a natural number? Answer
      
      
   2. Give three examples of natural numbers and three examples of numbers that are not natural numbers. Answer
      
      
2. Determine the pattern in each table and apply the pattern to complete the table.
   
   
   1. 
      

      1	2
      2	4
      3	6
      4
      5
      6

      
      Answer
      
      
   2. 
      

      −1	4
      2	7
      4	9
      9
      10
      15

      
      Answer
      
      
   3. 
      

      1	1
      2	4
      3	9
      4
      5
      6

      
      Answer
      
      
3. Rearrange the following formulas for the indicated variable.
   
   
   1. C = 2πr, r Answer
      
      
   2. P = 2 + 2w, w Answer
      
      
   3. b1 Answer
      
      
4. Determine the value for each of the formulas below by substituting the given variables.
   
   
    

   	Formula	Variables
   a.	V = wh	= 10 cm   w = 7 cm   h = 3 cm
   b.	vf = vi + at	vi = 20.34   a = 9.81   t = 5.22
   c.	A = 2πr2 + 2πrh	r = 4.5 in   h = 5.0 in

   
   
   Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

 

Visit Natural Numbers. Review the definition, and then follow the link to the demonstration applet. The demonstration applet shows how the number sets in the real number system are related to one another.

 

Move your cursor over the applet to see examples of natural numbers and examples from other number sets. Can you think of a few examples for each number set that are different than those shown in the applet?

 

You may need to ask your teacher for your school’s LearnAlberta login information.

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Use Extending Patterns Applet to review patterns in a table of values and pattern extension by applying the pattern rule. Try to discover how you can find patterns by comparing numbers up and down the same column and by comparing numbers across the same row.

 

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Now watch Exploring Patterns Video. Click on the tap-dancing image or the video link to watch an application of patterns in tap dancing!

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Try your hand at identifying and completing patterns by playing Exploring Patterns Video Interactive.

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Visit Formula for a definition and to review rearranging formulas. Under “Example” you will see how the width variable in the perimeter formula can be isolated by balancing both sides of the formula. There is a second example that shows how the radius variable in the circumference formula can be isolated.

 

Isolating variables is an important skill. This skill will save you a lot of time. You won’t need to guess and check, and you will have a method that lets you find the answer for any variable when given a formula.

Go back to the Are You Ready? questions and try them again. If you are still having difficulty, contact your teacher.

Discover

 

Try This 1

 

Digital Vision/Thinktock

 

Study the arrangement of marbles in each figure, and then answer the questions. Remember to save your responses in your course folder—you will need those responses in the upcoming Share 1.

 

1. If the pattern continues, how many marbles would each of the next two figures have? How did you determine your answer?
   
   
2. Describe how you would find the number of marbles in figure 20. Use your method from question 1 to determine the number of marbles.
   
   
3. Describe how you would find the number of marbles in figure 100. Use your method from question 1 or find a new method to determine the number of marbles.
   
   
4. Was your method for determining the number of marbles in figure 20 different from your method used for figure 100? If yes, why did you choose a different method? If no, can you think of another method that might be more efficient? Explain.

Save your work in your course folder.

 

Share 1

 

If you do not have access to a discussion board, another student, or an appropriate partner with whom to share your work, talk to your teacher. For more information on Share discussions, refer to the Course Introduction or contact your teacher.

 

Share your responses from Try This 1 with a partner or group.

• Compare the answers for questions 1, 2, and 3. Are the answers given by your classmates similar to your own? Explain reasons for any differences.
  
  
• Compare and discuss your method for solving questions 2 and 3 with methods used by your classmates.

If your teacher requires, you may need to place a summary of your Share discussion in your course folder.

 

You may have found that there really are a variety of ways to solve the problem. As you move through the lesson, you will encounter other strategies you can use when you revisit the Try This 1 problem later in the lesson.

Explore

 

Some parents teach their children how to manage their money by giving them an allowance, or pocket money. The amount of the allowance may depend on the child’s age, his or her responsibilities, and other factors. Have you ever received an allowance? How did you spend your money?

 

Suppose a parent decides to give her son $5 per week, starting on his 6th birthday. Every year, he receives a “raise” in his allowance—an additional $3 per week. Can you determine what his allowance will be when he turns 17? By the end of this lesson you will be able to figure that out.

 

 

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As you work through the lessons, you will come across math-related words that are unfamiliar to you. These words will be used over and over again, so it is important that you understand their meaning.

 

Save Module 1 Glossary Terms in your course folder now.

 

Each time you come across a new term in Mathematics 20-1, add the term and a definition to your copy of Module 1 Glossary Terms.

 

Remember that this glossary is for you! While your teacher will want to know that you are keeping your glossary document updated, the contents of the document will ultimately benefit you as you learn and review the concepts presented in this course. It is important that you write down definitions in a way that makes sense to you. For example, if you don’t understand the dictionary's definition, look elsewhere for wording that makes sense to you.

Try This 2

 

Consider these examples of sequences.

 

A	1, 8, 15, 22, 29, 36, ...
B	1, 2, 4, 8, 16, 32, ...
C	2, 4, 6, 8, 10, 12, 14, ...
D	2, 3, 5, 8, 12, 17, 23, ...
E	40, 35, 30, 25, 20, 15, ...

1. In your own words, describe what is meant by a sequence.

2. Some of the sequences in the table are based on a similar pattern. Study the sequences. Can you identify which sequences have a similar pattern?

3. How are the sequences that you selected similar? Describe the common pattern.

4. Refer to the definition for arithmetic sequence that you added to Module 1 Glossary Terms. Determine which sequences in the examples given in the table are arithmetic sequences. Were these the same sequences you selected in question 2?

5. Look through the six sequences again to make sure you didn’t miss any. Describe the properties of the other sequences that disqualify them as arithmetic sequences.

Save your responses in your course folder.

Arithmetic Sequences Increase by a Specific Factor

 

In Try This 2 you may have recognized that an arithmetic sequence increases by a specific factor. Example 1 in the image shows an arithmetic sequence that increases by a specific factor, 3, as you move between terms. An arithmetic sequence can also have terms that decrease by a factor (e.g., −5), as shown in example 2.

 

You will now explore methods used to find any term in a sequence. In the first method you will repeat many calculations on your calculator. In the second method you will apply a formula.

 

Finding a Term in an Arithmetic Sequence

 

Calculator Method

 

In Try This 1 you came up with a method to find the 20th number in the arithmetic sequence 10, 14, 18, ….

 

You may have started with the first term, and then used a calculator to repeatedly add the number 4, which is the common difference in this sequence.

 

Many calculators can simplify repeated calculations.

 

Follow these steps to find the 20th term of the sequence 10, 14, 18, ….

 

Step 1: Press , the first term in the sequence.

 

Step 2: Press then then . The result will be the second term in the sequence.

 

Step 3: While counting the terms, continue to press until you obtain the value of the 20th term. You do not need to press then repeatedly.

 

Try This 3

1. Use the calculator method to find the 100th term in 10, 14, 18, ….

2. What do you think are the advantages and disadvantages of evaluating repeated calculations with a calculator?

Place a copy of your responses in your course folder.

Algebraic Method

 

Even with a calculator, repeated calculations may not be very efficient, especially if you were asked to determine the value of a term that is far into the sequence. That’s where an algebraic method helps.

 

You can use an algebraic method to identify specific terms in an arithmetic sequence. In Try This 4 you will develop a formula for evaluating arithmetic sequences.

 

When working with arithmetic sequences, you will encounter and use terms that describe the sequence’s properties. These variables include the following:

 

Variable	Meaning
a	the first term in the sequence (or t1)
d	the common difference
n	the position of the term in the sequence
tn	the value of the nth term

 

You can use the variable tn to refer to any term in the sequence. Here are some examples:

• If you are solving for the third term in a sequence, simply replace the n in tn with the number 3. So, t3.
  
  
• If you are solving for the 10th term, simply replace the n in tn with the number 10. So, t10.

If you are solving for the 150th term, what variable could you use to refer to this term?

Try This 4

 

Now that you are familiar with the variables that will be used, you are ready to construct a formula. You will do this in two different ways. After constructing your formula, you will have an opportunity to share your discoveries and your strategies with others.

 

Part 1

1. Complete a table like Table 1 by simplifying each of the formulas. Record your answers under the Simplified Form column.
   
   
    

   TABLE 1: FORMULAS DESCRIBING SEQUENCES IN TABLE 1
   	Original Formula	Simplified Form
   1	tn = −29 + (n − 1)(7)
   2	tn = 2 + (n − 1)(3)
   3	tn = n − 1
   4	tn = 96 + (n − 1)(−10)

   
   
   
2. Match each sequence in Table 2 with its corresponding formula from Table 1. For example, if you think sequence A is described by formula 3, then you would write “A3” in your copy of Table 2.
   
   
    

   TABLE 2: EXAMPLES OF ARITHMETIC SEQUENCES
   	Sequence	Formula from Table 1
   A	0, 1, 2, 3, ...
   B	2, 5, 8, 11, ...
   C	96, 86, 76, 66, ...
   D	−29, −22, −15, −8, ...

   
   
   
3. What strategies did you use to determine how the formulas matched the sequences?
   
   
4. What patterns do you see between the original formulas and the properties of sequences?
   
   
5. Based on the pattern you observed, what would the formula to describe the following sequence look like?
   
   
    
   23, 31, 39, 47, 55, ...
   
   
6. Study the relationship between the sequences and their corresponding simplified formulas (found in the last column of Table 2). How could you construct a formula in the simplified form without writing the formula in the original form?

Part 2

 

In Part 2, consider an arithmetic sequence that begins with the term a and has a common difference d.

 

Watch Arithmetic Sequence Video for an explanation of the table in question 7.

7. Complete the table by writing an expression for t1, t2, t3, t4, t5, and t6 in terms of a and d. Simplify the expression. The first three rows have been completed for you.
   
   
    

   n	tn	Expression
   1	t1	a
   2	t2	a + d
   3	t3	a + d + d or a + 2d

   
   
   
8. Describe the relationship between n and the coefficient of d in the expression. Express the relationship using the variable n.
   
   
9. Extend the pattern you noticed in the table. What would be the expression for t20?
   
   
10. What would be the expression for tn?

Save your results in your course folder.

 

General Term of the Sequence

 

From working through Try This 4, you may have discovered that you can find the value of any term in an arithmetic sequence by using the formula tn = a + (n − 1)d.

 

This formula is called the general term of the sequence. As with any formula, if you are given values for all but one variable, you will be able to solve the equation for the unknown variable. In the remainder of the lesson you will use the general term to determine tn, a, n, and d.

Example 3

 

Work through Dakota’s Example. The general term of the sequence formula will be used to help solve the problem. You might recognize the question from the beginning of Explore.

Self-Check 2

 

Assume that, at age 17, Dakota gets a part-time job earning $12/h. He receives a signing bonus of $250 when he begins his job.

 

Stockbyte/Thinkstock

1. Complete the table relating Dakota's earnings to the number of hours worked.
   
   
    

   Number of Hours Worked	Earnings ($)
   0	250
   1	262
   2
   3
   4
   5

   
   
   

   Answer

Connect

 

© Johnny Lye/123827/Fotolia

 

Lesson 1 Assignment

Lesson 1 Summary

 

In this lesson you investigated the following questions:

• How can you find any term in an arithmetic sequence?

• How are patterns important in the real world?

You studied number patterns called sequences. Specifically, you learned about the properties of arithmetic sequences, You derived a formula, called the general term, that can be used to represent any arithmetic sequence and determine its properties.

 

You relied on your ability to discover patterns and relationships among numbers. Finding and using patterns is important in mathematics. Many of the formulas you see in math are descriptions of patterns.

 

Patterns are also important in daily living. Accurate weather forecasting, for example, is something people depend on to plan their business and recreational activities. In the entertainment industry, television series are cancelled, extended, or moved to other time slots based on audience viewing patterns.

 

In the next lesson you will investigate where arithmetic sequences are applied in real-world contexts. You will also discover how arithmetic sequences are related to linear functions.