Lesson 4

Lesson 4: Geometric Sequences

Focus

Do you have your driver’s license? For many people, passing their road test is a huge milestone and another step towards freedom and independence. For many parents, another driver in the family means they can share the responsibility of driving with their children. A new car, however, isn’t cheap. On top of that, there are insurance, fuel, and maintenance costs to owning a vehicle. Nevertheless, buying a car can be a thrilling experience, especially when it’s your first car!

Many people borrow money to buy a vehicle. This is called financing the vehicle. The benefit of financing is that you get to use the vehicle right away. The drawback is that you have to pay the interest on the outstanding balance after each month.

To keep things predictable for the borrower, lenders use the mathematics of sequences to determine monthly payments.

In this lesson you will investigate geometric sequences. Just as you did with arithmetic sequences, you will use discovered patterns to derive rules for determining the general term of a geometric sequence. You will also explore problems, such as those involving compound interest, that can be modelled by geometric sequences.

Outcomes

At the end of this lesson you will be able to

• identify assumptions made when identifying a geometric sequence
  
  
• provide and justify an example of a geometric sequence
  
  
• derive a rule for determining the general term of a geometric sequence
  
  
• solve a problem that involves a geometric sequence

Lesson Questions

You will investigate the following questions:

• How are arithmetic sequences and geometric sequences similar and different?
  
  
• How is compound interest an application of geometric sequences?

Assessment

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

• completion of the Lesson 4 Assignment (Download the Lesson 4 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

Materials and Equipment

You will need a sheet of paper.

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Are You Ready?

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

1.  
   1. What is 40% of 200? Answer
      
      
   2. 15 is 30% of what number? Answer
      
      
   3. An investment of $25 000 earns 7% interest per year. What is the simple interest earned on this investment after 1 yr? Answer
      
      
2. What is a sequence? Answer
   
   
3. For each arithmetic sequence, determine the general term and the value of t11.
   
   
   1. −9, −2, 5, 12, … Answer
      
      
   2. 400, 365, 330, 295, … Answer
      
      
4. What is the order of operations? Answer
   
   
5. Evaluate each of the following expressions.
   
   
   1. 3 + 2 × 5 Answer
      
      
   2. 300 × (5 − 1)2 Answer
      
      
   3. Answer

How did the questions go? Depending on your answer, skip forward to Discover or work through Refresher.

Refresher

Use Math Continuum: Percent (Food) to review concepts relating to percentage. In the first topic you will review the definition of percent. In the second topic you will practise determining equivalent percents, fractions, and decimals. In the final topic, you will play a game that tests your knowledge of percents and your reflexes!

To get started, select “Percent (Food)” under the “Number” column. Then start with the first topic.

Next, look at the definition in Sequence. You may want to retrieve your copy of Module 1 Glossary Terms to update your definition of sequence.

Review the basic properties of arithmetic sequences by revisiting Direct and Partial Variation: Arithmetic Sequences. This applet also reviews the relationship between arithmetic sequences and linear functions. Work through slides 1 to 11. Remember that any linear function with a y-intercept of zero exhibits direct variation, and any linear function with a non-zero y-intercept represents partial variation.

Use Order of Operations to remind yourself how to evaluate expressions that involve multiple mathematical operations. The applet includes three examples of varying difficulty.

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

Discover

How many times can you fold a piece of paper in half? You will answer that question in this Math Lab. You will also learn something about another type of sequence.

Explore

A pyramid scheme is a fraudulent money-making scam. These schemes are illegal because they deceive people by promising huge sums of money. The idea of a pyramid scheme is based on a person or group of people recruiting other people. Those who have been recruited pay a fee to the recruiter.

The recruited individuals, in turn, recruit more people who also pay a fee. The fees are distributed to those who are higher up in the pyramid. To make money in this scheme, you must have many levels of recruited individuals beneath you.

The illustration shows an example of how a pyramid scheme works. In this example each recruited person recruits six other people. Do you think this practice can continue on forever?

The number of people recruited at each level forms a sequence. In this lesson you will investigate sequences of this kind. You will discover whether a pyramid scheme is sustainable. You will determine the general term for these sequences, and you will use the general term to solve problems related to finance and other applications.

Try This 1

1. Work through Examples of Geometric Sequences. Find the common ratio, and then use the common ratio to extend each sequence.
   
   
    
   
   
   
   
2. Consider the following sequences.
   
   
    

   A	3, 15, 75, 375, 1875, …
   B	8, 12, 18, 27, …
   C	−5, −10, 20, 40, −80, −160, …
   D	1215, 405, 135, 45, …,
   E	2, 4, 8, 16, 32, 64, 128, 254, 508

   
   
   
   1. How are the first three sequences different?
      
      
   2. Which of the sequences are geometric? How do you know?

Challenging Assumptions

Not all sequences that appear to be geometric are, in fact, geometric. This lesson has assumed that if the first few terms of an infinite sequence show a geometric pattern, the rest of the sequence is also geometric.

Generally, the more terms a sequence shows, the safer it is to assume the sequence has a particular pattern. On the other hand, if only a few terms are shown, there may be other possible patterns governing the progression.

How many terms must be revealed in a sequence before you can confidently identify the sequence as geometric? Think about this as you complete Try This 2.

Try This 2

1. Consider the sequence 1, 2, …. What are some possible values for the next term in the sequence? State two possibilities and describe the reasons behind your choices.
   
   
2. Consider the sequence 1, 2, 4, …. Are there enough terms shown that you can be certain this is a geometric sequence? Can you find any other pattern in these three terms that would not result in 8 as the 4th term?

Save your answers in your course folder.

You will explore this concept further if you choose to complete Going Beyond later in this lesson.

Self-Check 1

Check your understanding by completing the following interactive activities:

• Create-a-Sequence
  
  
   
  
  
  
  
• Combining Sequences
  
  
   
  
  
  
  
• Identifying the Common Ratio
  
  
   
  
  
  

Try This 3

Retrieve your work from the Math Lab. You may have found that you can only fold a sheet of paper fewer than 10 times. Assume that you can fold a piece of paper indefinitely. You could then use the pattern to determine a formula for the general term of a geometric sequence.

1. Copy the information from the “Layers” column of your table into the second column of the following table. The first three rows have been done as an example. Complete all rows in your table.
   
   
    

   Term Position	Term Value	Expression
   t1	1	1
   t2	2	1 × 2
   t3	4	1 × 2 × 2

   
   
   
2. Now you can repeat the same exercise in the general case. Create and complete a second table, such as the one shown here. Assume that the first term is a, and the common ratio is r. Rewrite the expressions from the table in question 1 using a and r. The first three rows have been completed for you.
   
   
    

   Term Position	Expression
   t1	a
   t2	a × r = ar
   t3	a × r × r = ar2

   
   
   

3. Compare the term position in the sequence and the expression. What pattern do you see?
   
   

4. Use the pattern you identified to determine an expression for the general term tn.

Share 1

Work with a classmate or in a group.

1. Compare your expression for the general term tn from Try This 3. If the expressions vary, discuss reasons for the differences and try to agree on the correct expression. You may want to search the textbook or the Internet for a formula for the general term of a geometric sequence.
   
   
2. Use your expression to build the general term that could be used to describe the geometric sequence representing the number of people in each level of the pyramid scheme introduced at the beginning of Explore.
   
   
    
   
   
   
3. Consider the sequence 24, 36, 54, 81, ….
   
   
   1. How do you know this is a geometric sequence? What assumption are you making?
      
      
   2. What is the common ratio?
      
      
   3. What is the general term that describes the sequence?

Save your work in your course folder. Ask your teacher if you should submit your work for feedback.

The General Term of a Geometric Sequence

Recall that the general term is the formula that describes the terms of a sequence. In Try This 3, you may have discovered that the general term of a geometric sequence is as follows:

tn = arn−1

As a rule, you can use the general term to solve problems related to specific terms of a geometric sequence.

Try This 4

The world’s largest free-standing totem pole is located in Beacon Hill Park in Victoria, British Columbia. The pole is approximately 128 ft tall.

Imagine that an eagle with a golf ball clasped in its talons is perched at the top of the totem pole. As it flies away, the eagle drops the ball. The ball bounces to 65% of its previous height.

Figure 1

Assume that the ball continues to bounce to 65% of its previous height.

1. Calculate and label the first four bounce heights shown in Figure 1.

2. The bounce heights in this problem form a sequence. Write the general term for the sequence.

3. Use the general term to find the height the ball reaches after the 9th bounce.

Save your work in your course folder.

Compound Interest

Recall that a pyramid scheme is a fraudulent money-making scam that cheats many people out of their money. Despite the existence of pyramid schemes and other scams, there are a large number of legitimate ways to invest your money. For example, you may invest your money in a savings account, Guaranteed Investment Certificate (GIC), or Canada Savings Bond to name a few. These investments pay compound interest.

In Try This 3 and Share 1 you derived the general term of a geometric sequence. Now you can derive the formula for evaluating compound interest. As you watch Deriving a Formula for Compound Interest, think about how you would answer the following questions:

1. How is compound interest an example of a geometric sequence? What is the first term? What is the common ratio?
   
   
2. How was the process of deriving the formula for compound interest similar to the process for deriving the general term of a geometric sequence?
   
   
3. How would you use the formula to determine the following?
   
   
   1. P
      
      
   2. i
      
      
4. How is the formula A = 800(1.1)n different from the general term tn = 800(1.1)n−1?

Share 2

With a classmate, answer the four questions you thought about as you watched Deriving a Formula for Compound Interest. Record your solutions and show support for your reasoning.

Save your work in your course folder.

Your teacher may look at your work to gauge your progress and provide feedback.

Try This 5

Go to your course folder and retrieve your work from Lesson 2 Try This 1. In that exercise you investigated the investment of $1000 in a savings bond that pays 4% simple interest at the end of every year. The investment earned $40/yr. The value of the investment at the end of each year formed an arithmetic sequence.

In the video Deriving a Formula for Compound Interest, you solved a problem involving the investment of $1000 in a savings bond that pays 4%/yr compounded annually. The value of the investment at the end of each year formed a geometric sequence.

1. In what ways are these problems similar? In what ways are they different?

2. In Lesson 2 Try This 2, you created a graph relating the value of the investment over the first 10 yr. Create a new graph relating the value of the compound-interest investment over the first 10 yr.

3. What conclusions can you draw about compound interest as compared to simple interest?

4. Which investment, the investment with simple interest or the investment with compound interest, is worth more at the end of 10 yr? By approximately how much?

Connect

Lesson 4 Assignment

Lesson 4 Summary

In this lesson you investigated the following questions:

• How are arithmetic sequences and geometric sequences similar and different?

• How is compound interest an application of geometric sequences?

You studied geometric sequences. The following table summarizes the similarities and differences between arithmetic and geometric sequences.

Geometric sequences can be applied in many situations. One such situation is compound interest. You compared the formula for compound interest, A = P(1 + i)n, to the general term. In the case of compound interest, the first term is the principal and the common ratio is 1 + i.

You also compared simple-interest growth to compound-interest growth. Over time, an investment earning compound interest increases in value more rapidly than the same investment earning simple interest.