Understanding Geometric Sequences
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1 6 Locker LESSON 1.1 Understanding Geometric Sequences Texas Math Standards The student is expected to: A1.1.D Write a formula for the nth term of arithmetic and geometric sequences, given the value of several of their terms. Mathematical Processes A1.1.F Analyze mathematical relationships to connect and communicate mathematical ideas. Language Objective 3.F, 3.H, 4.G Explain to a partner how to tell whether a sequence is a geometric sequence. Name Class Date 1.1 Understanding Geometric Sequences Essential Question: How are the terms of a geometric sequence related? A1.1.D write a formula for the nth term of geometric sequences, given the value of several of their terms. Explore 1 Exploring Growth Patterns of Geometric Sequences The sequence 3, 6, 1, 4, 48, is a geometric sequence. In a geometric sequence, the ratio of successive terms is constant. The constant ratio is called the common ratio, often represented by r. Complete each division. 6 3 = 1 6 = 4 1 = 48 4 = The common ratio r for the sequence is. Use the common ratio you found to identify the next term in the geometric sequence. The next term is 48 = 96. Reflect Resource Locker ENGAGE Essential Question: How area the terms of a geometric sequence related? The terms of a geometric sequence are related by a common ratio, often represented by r. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and examples of payment plans students might use when charging for odd jobs. Then preview the Lesson Performance Task. 1. Suppose you know the twelfth term in a geometric sequence. What do you need to know to find the thirteenth term? How would you use that information to find the thirteenth term? You need to know the common ratio, r. You can multiply the twelfth term by r.. Discussion Suppose you know only that 8 and 18 are terms of a geometric sequence. Can you find the term that follows 18? If so, what is it? Only if you know that 8 and18 are successive terms. In that case, the common ratio is 16, and the next term is 048. However, 8 and 18 could be terms of a different geometric sequence. For example, in the geometric sequence 8, 16, 3, 64, 18,..., the next term is 56. Explore Comparing Growth Patterns of Arithmetic and Geometric Sequences Recall that in arithmetic sequences, successive (or consecutive) terms differ by the same nonzero number d, called the common difference. In geometric sequences, the ratio r of successive terms is constant. In this Explore, you will examine how the growth patterns in arithmetic and geometric sequences compare. In particular, you will look at the arithmetic sequence 3, 5, 7,... and the geometric sequence 3, 6, 1,.... Module Lesson 1 Name Class Date 1.1 Understanding Geometric Sequences Essential Question: How are the terms of a geometric sequence related? A1.1.D write a formula for the nth term of geometric sequences, given the value of several of their terms. Explore 1 Exploring Growth Patterns of Geometric Sequences Resource The sequence 3, 6, 1, 4, 48, is a geometric sequence. In a geometric sequence, the ratio of successive terms is constant. The constant ratio is called the common ratio, often represented by r. Complete each division. 3 = 1 6 = 4 1 = 48 4 = The common ratio r for the sequence is. Use the common ratio you found to identify the next term in the geometric sequence. The next term is 48 =. Reflect Suppose you know the twelfth term in a geometric sequence. What do you need to know to find the thirteenth term? How would you use that information to find the thirteenth term? You need to know the common ratio, r. You can multiply the twelfth term by r.. Discussion Suppose you know only that 8 and 18 are terms of a geometric sequence. Can you find the term that follows 18? If so, what is it? Only if you know that 8 and18 are successive terms. In that case, the common ratio is 16, and the next term is 048. However, 8 and 18 could be terms of a different geometric sequence. For example, in the geometric sequence 8, 16, 3, 64, 18,..., the next term is 56. Explore Comparing Growth Patterns of Arithmetic and Geometric Sequences Recall that in arithmetic sequences, successive (or consecutive) terms differ by the same nonzero number d, called the common difference. In geometric sequences, the ratio r of successive terms is constant. In this Explore, you will examine how the growth patterns in arithmetic and geometric sequences compare. In particular, you will look at the arithmetic sequence 3, 5, 7,... and the geometric sequence 3, 6, 1,.... Module Lesson 1 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 545 Lesson 1.1
2 The tables shows the two sequences. A B 3, 5, 7,... 3, 6, 1,... Term Number Term Term Number Term The common difference d of the arithmetic sequence is 5-3 =. The common ratio r of the geometric sequence is 6 =. 3 Complete the table. Find the differences of successive terms. Arithmetic: 3, 5, 7,... Term Number Term Difference = = = = Geometric: 3, 6, 1,... Term Number Term Difference = = = = 4 Module Lesson 1 PROFESSIONAL DEVELOPMENT Learning Progressions In an earlier module, students studied arithmetic sequences and wrote general recursive and explicit rules for them. Students used these rules to solve real-world problems involving arithmetic sequences. In this module, students will learn about geometric sequences and exponential functions. In the next module, students will learn more about exponential functions, including exponential growth and decay functions. EXPLORE 1 Exploring Growth Patterns of Geometric Sequences INTEGRATE TECHNOLOGY Have students complete the Explore activity in either the book or online lesson. CONNECT VOCABULARY Make sure that students understand the meanings of successive terms and ratio of successive terms. You can explain that two successive terms are two terms that are next to each other in the sequence. Have students give examples of pairs of successive terms. Explain that successive terms can also be called consecutive terms. EXPLORE Comparing Growth Patterns of Arithmetic and Geometric Sequences QUESTIONING STRATEGIES How are the graphs of geometric sequences and arithmetic sequences alike? How are they different? Possible answer: They can both be represented by a function with a domain that is the set of positive integers, or a subset of consecutive positive integers beginning with 1. The graph of a geometric sequence follows a curve, while the graph of an arithmetic sequence is linear. Understanding Geometric Sequences 546
3 INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Show students a graph of the first 4 terms of f (n) = + (n - 1) and f (n) = n - 1. Point out that the shape formed by the points indicates whether the graph represents an arithmetic or geometric sequence. Look at three or more points before determining whether the graph is linear or exponential, since arithmetic and geometric sequences can have the same first two terms. EXPLAIN 1 Extending Geometric Sequences AVOID COMMON ERRORS When finding a common ratio, students might divide in the wrong order. Tell them they are really finding what each term is being multiplied by to get the next term, so the inverse operation (division), will produce r. For example, when finding the common ratio of the geometric sequence 16, 4, 1, 1 4,..., divide 4 by 16. C D Compare the growth patterns of the sequences based on the tables. For the arithmetic sequence, the differences are equal. The terms of the arithmetic sequence increase by a fixed amount. For the geometric sequence, the differences increase. The terms of the geometric sequence increase by an increasing amount. Graph both sequences in the same coordinate plane. Compare the growth patterns based on the graphs. Reflect Term y Sequence Patterns Geometric sequence Arithmetic sequence Term number For the arithmetic sequence, the slope is constant, so the vertical distances between successive points are the same. For the geometric sequence, the vertical distances between the points increase by increasing amounts. 3. Which grows more quickly, the arithmetic sequence or the geometric sequence? The geometric sequence Explain 1 Extending Geometric Sequences In Explore 1, you saw that each term of a geometric sequence is the product of the preceding term and the common ratio. Given terms of a geometric sequence, you can use this relationship to write additional terms of the sequence. Finding a Term of a Geometric Sequence For n, the nth term, ƒ (n), of a geometric sequence with common ratio r is ƒ (n) = ƒ (n - 1) r. x Module Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Have one student write and graph a geometric sequence for which the common ratio r is greater than 1. Have the other student write and graph a geometric sequence for which the common ratio r is 0 < r < 1. Have the students compare the graphs. Have the students switch roles and repeat the exercise. 547 Lesson 1.1
4 Example 1 Find the common ratio r for each geometric sequence and use r to find the next three terms. 6, 1, 4, 48, 1 =, so the common ratio r is. 6 For this sequence, ƒ (1) = 6, ƒ () = 1, ƒ (3) = 4, and ƒ (4) = 48. ƒ (4) = 48, so ƒ (5) = 48 () = 96. ƒ (5) = 96, so ƒ (6) = 96 () = 19. QUESTIONING STRATEGIES In giving the formula for the nth term of a geometric sequence, why does the book say For n? It is understood that n is an integer. The first term of the sequence is f (1). For n = 0 and 1, f (n - 1) is not defined. ƒ (6) = 19, so ƒ (7) = 19 () = , 19, and , 50, 5, 1.5, 50 = 0.5, so the common ratio r is For this sequence, ƒ (1) = 100, ƒ () = 50, ƒ (3) = 5, and ƒ (4) = 1.5. ƒ (4) = 1.5, so ƒ (5) = 1.5 (0.5) = 6.5. ƒ (5) = 6.5, so ƒ (6) = 6.5 (0.5) = ƒ (6) = 3.15, so ƒ (7) = 3.15 (0.5) = , 3.15, and Reflect 4. Communicate Mathematical Ideas A geometric sequence has a common ratio of 3. The 4th term is 54. What is the 5th term? What is the 3rd term? The 5th term is 3 times the 4th, or 16. The 4th term, 54, is 3 times the 3rd term, so the 3rd term is 18. Your Turn Find the common ratio r for each geometric sequence and use r to find the next three terms. 5. 5, 0, 80, 30, , -3, 1, - 3, = 4 = r -3 9 = - 3 = r f (4) = 30, so f (5) = 30 (4) = 180. f (4) = - 3, so f (5) = - 3 ( - 3 ) = 9. f (5) = 180, so f (6) = 180 (4) = 510. f (5) = f (6) = 510, so f (7) = 510 (4) = 0,480. 9, so f (6) = 9 ( - 3 ) = - 7. The next three terms are 180, 510, f (6) = - and 0,480. 7, so f (7) = - 7 ( - 3 ) = 81. The next three terms are 9, - 7, and 81. Module Lesson 1 DIFFERENTIATE INSTRUCTION Multiple Representations Help students making connections between a table representing an arithmetic sequence and a graph of the sequence. For example, start with the table and have students make the graph. Then start with the graph and have students make the table. Do the same for a geometric sequence. Have students compare the graphs of the two sequences. Understanding Geometric Sequences 548
5 EXPLAIN Recognizing Growth Patterns of Geometric Sequences in Context QUESTIONING STRATEGIES If you know that a sequence is a geometric sequence, how can you find the common ratio? Choose any term after the first term and divide it by the preceding term. The result is the common ratio. Explain Recognizing Growth Patterns of Geometric Sequences in Context You can find a term of a sequence by repeatedly multiplying the first term by the common ratio. Example A bungee jumper jumps from a bridge. The table shows the bungee jumper s height above the ground at the top of each bounce. The heights form a geometric sequence. What is the bungee jumper s height at the top of the 5th bounce? Bounce Height (feet) Find r = 0.4 = r ƒ (1) = 00 ƒ () = 80 First bounce 00 ft Second bounce 80 ft Third bounce 3 ft 1 = 00 (0.4) or 00 (0.4) ƒ (3) = 3 = 80 (0.4) = 00 (0.4) (0.4) = 00 (0.4) In each case, to get ƒ (n), you multiply 00 by the common ratio, 0.4, n -1 times. That is, you multiply 00 by (0.4) n - 1. The jumper s height on the 5th bounce is ƒ (5). Multiply 00 by (0.4) 5-1 = (0.4) (0.4) 4 = 00 (0.056) = 5.1 The height of the jumper at the top of the 5th bounce is 5.1 feet. Module Lesson 1 LANGUAGE SUPPORT Connect Vocabulary Remind students that in a geometric sequence, the ratio of successive terms is constant. This constant ratio is called the common ratio, often written as r. Point out that it is called a common ratio because it is shared by all the pairs of successive terms. Discuss other uses of this meaning of the word common; for example, a common boys name. 549 Lesson 1.1
6 Example A ball is dropped from a height of 144 inches. Its height on the 1st bounce is 7 inches. On the nd and 3rd bounces, the height of the ball is 36 inches and 18 inches, respectively. The heights form a geometric sequence. What is the height of the ball on the 6th bounce to the nearest tenth of an inch? Find r. HOME CONNECTION Have students think of a real-world situation that can be represented by a geometric sequence = = r ƒ (1) = 7 ƒ () = 36 = 7 ( ) or 7 ( ) ƒ (3) = 18 = 36 ( ) = 7 ( )( ) = 7 ( ) 1 In each case, to get f(n), you multiply 7 by the common ratio,, n - 1 times. That is, you ( ) n - 1 multiply 7 by. The height of the ball on the 6th bounce is f ( ). ( ( Multiply 7 by ) = ). 7 ( ) 5 = 7 ( 3 ) = 7 (0.315) The height of the ball at the top of the 6th bounce is about Reflect =.5 6 inches. 7. Is it possible for a sequence that describes the bounce height of a ball to have a common ratio greater than 1? No; if the common ratio were greater than 1, the bounce height would increase, which could not happen in the real world..3 Module Lesson 1 Understanding Geometric Sequences 550
7 ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Use examples to help students make generalizations about geometric sequences of positive numbers for which the common ratio r is greater than 1 and geometric sequences of positive numbers for which the common ratio r is between 0 and 1, that is, 0 < r < 1. Your Turn 8. Physical Science A ball is dropped from a height of 8 meters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 4 th bounce? Round your answer to the nearest tenth of a meter. Bounce Height (m) = 0.75 = r f (4) = 6 (0.75) 4-1 f (4) = 6 (0.75) 3 =.5315 The ball bounces about.5 meters on the 4th bounce. SUMMARIZE THE LESSON Complete the graphic organizer with students to discuss and summarize lesson concepts. In each box, write a way to represent the geometric sequence. Elaborate 9. Suppose all the terms of a geometric sequence are positive, and the common ratio r is between 0 and 1. Is the sequence increasing or decreasing? Explain. If r is between 0 and 1 and all the terms are positive, then each term is less than the preceding term. So, the sequence is decreasing. Table Ways to Represent the Geometric Sequence 1,, 4, 8, Term Number Term n -1 Formula f (n) = 1 () Words Start with 1 and multiply each term by to get the next term. 10. Essential Question Check-In If the common ratio of a geometric sequence is less than 0, what do you know about the signs of the terms of the sequence? Explain. The signs of the terms alternate. If r < 0 and the first term is negative, the second is positive. Then the third must be negative. The signs of the terms continue to alternate. If the first term is positive, the results are similar. Module Lesson Lesson 1.1
8 Evaluate: Homework and Practice Find the common ratio r for each geometric sequence and use r to find the next three terms. Online Homework Hints and Help Extra Practice EVALUATE 1. 5, 15, 45, , 6, -18, = 3 = r - = -3 = r f (4) = 135, so f (5) = 135 (3) = 405. f (4) = 54, so f (5) = 54 (-3) = -16. f (6) = 405 (3) = 115 f (7) = 115 (3) = , 115, and , 0, 100, 500, 4. 8, 4,, 1, = 5 = r 8 = = r f (4) = 500, so f (5) = 500 (5) = 500. f (6) = 500 (5) = 1,500 f (7) = 1,500 (5) = 6, , 1,500, and 6,500. f (6) = -16 (-3) = 486 f (7) = 486 (-3) = , 486, and f (4) = 1, so f (5) = 1 ( ) = f (6) = ( ) = 4 f (7) = 4 ( ) = , -36, 18, -9, 6. 00, -80, 3, -1.8, = - = r -80 f (4) = -9, so f (5) = -9 ( - ) = 4.5. f (6) = 4.5 ( - ) = -.5 f (7) = -.5 ( - ) = , -.5, and ,, and = -0.4 = r f (4) = -1.8, so f (5) = -1.8 (-0.4) = 5.1. f (6) = 5.1 (-0.4) = f (7) = (-0.4) = , -.048, and , 30, 90, 70, 8. 5, 3, 1.8, 1.08, = 3 = r 5 = 0.6 = r f (4) = 70, so f (5) = 70 (3) = 810. f (4) = 1.08, so f (5) = 1.08 (0.6) = f (6) = 810 (3) = 430 f (7) = 430 (3) = , 430, and 790. f (6) = (0.6) = f (7) = (0.6) = , , and ASSIGNMENT GUIDE Concepts and Skills Explore 1 Exploring Growth Patterns of Geometric Sequences Explore Comparing Growth Patterns of Arithmetic and Geometric Sequences Example 1 Extending Geometric Sequences Example Recognizing Growth Patterns of Geometric Sequences in Context Practice Exercise Exercise 3 Exercises 1 14, 1 Exercises 15 0 Module 1 55 Lesson 1 Exercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall 1.D Multiple Representations Skills/Concepts 1.C Select Tools Recall 1.C Select Tools 1 Skills/Concepts 1.F Analyze Relationships 3 Strategic Thinking 1.C Select Tools 3 3 Strategic Thinking 1.G Explain and Justify Arguments Understanding Geometric Sequences 55
9 AVOID COMMON ERRORS When finding a common ratio, students might divide in the wrong order. Make sure that students divide each term after the first term by the preceding term to find the common ratio , 36, 7, , 16, 108, 7, = = r 43 = 3 = r f (4) = 144, so f (5) = 144 () = 88 f (6) = 88 () = 576 f (7) = 576 () = , 576, and 115. f (4) = 7, so f (5) = 7 ( 3 ) = 48. f (6) = 48 ( 3 ) = 3 f (7) = 3 ( 3 ) = , 3, and 1 3. Find the indicated term of each sequence by repeatedly multiplying the first term by the common ratio. Use a calculator , 8, 64, ; 5th term 1. 16, -3., 0.64, ; 7th term 8 1 = 8 = r = -0. = r f (5) = 1 (8) 5-1 f (7) = 16 (-0.) 7-1 = 1 (8) 4 = 4096 = 16 (-0.) 6 = , 15, -4.5, ; 5th term 14. 3, -1, 48, ; 6th term = -0.3 = r -1 3 = -4 = r f (5) = -50(-0.3) 5-1 f (6) = 3 (-4) 6-1 = -50 (-0.3) 4 = = 3 (-4) 5 = -307 Solve. You may use a calculator and round your answer to the nearest tenth of a unit if necessary. 15. Physical Science A ball is dropped from a height of 900 centimeters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 5th bounce? Bounce Height (cm) = 0.7 = r Find the 5th term of the sequence. f (5) = 800 (0.7) 5-1 = 800 (0.7) 4 = centimeters Module Lesson Lesson 1.1
10 16. Leo s bank balances at the end of months 1,, and 3 are $1500, $1530, and $ , respectively. The balances form a geometric sequence. What will Leo s balance be after 9 months? The first two terms of the sequence are 1500 and = 1.0 = r Find the 9th term of the sequence. f (9) = 1500 (1.0) 9-1 = 1500 (1.0) 8 = $ CRITICAL THINKING Have students explain why some sequences alternate signs and some do not. (When the common ratio, r, is negative, the terms in a geometric sequence alternate signs. When the common ratio is positive, the terms are all positive or all negative.) 17. Biology A biologist studying ants started on day 1 with a population of 1500 ants. On day, there were 3000 ants, and on day 3, there were 6000 ants. The increase in an ant population can be represented using a geometric sequence. What is the ant population on day 5? = = r Find the 5th term of the sequence. f (5) = = = 4,000 ants 18. Physical Science A ball is dropped from a height of 65 centimeters. The table shows the height of each bounce. The heights form a geometric sequence. How high does the ball bounce on the 8th bounce? = 0.8 = r Find the 8th term of the sequence. f (8) = 500 (0.8) 8-1 = 500 (0.8) 7 = centimeters. 19. Finance The table shows the balance in an investment account after each month. The balances form a geometric sequence. What is the amount in the account after month 6? = 1. = r Find the 6th term of the sequence. f (6) = 1700 (1.) 6-1 = 1700 (1.) 5 Bounce Height (cm) Month Amount ($) = $ Module Lesson 1 Image Credits: Jolanta Dabrowska/Alamy Understanding Geometric Sequences 554
11 JOURNAL Have students determine whether each sequence is arithmetic, geometric, or neither, and explain their reasoning: 3, 6, 9, 1, 15, arithmetic 3, 6, 10, 15, 1, neither 3, 6, 1, 4, 48, geometric 0. Biology A turtle population grows in a manner that can be represented by a geometric sequence. Given the table of values, determine the turtle population after 6 years. Year Number of Turtles = 3 = r Find the 6th term of the sequence. f (6) = = ; 115 turtles 1. Consider the geometric sequence 8, 16, 3,... Which of the following statements are true? a. The common ratio is. 16 = -, so the common ratio is b. The 5th term of the sequence is 18. The common ratio is -, and the 1st term is -8, f (n) = -8 (-) n-1, and f (5) = -8 (-) 4 = -18, and the 5th term of the sequence is -18. Image Credits: Olha K/ Shutterstock c. The 7th term is 4 times the 5th term. The 7th term is - times the 6th term, which is - times the 5th term, so the 7th term is 4 times the 5th term. d. The 8th term is 104. The 8th term is -8 (-) 7 = 104. e. The 10th term is greater than the 9th term. -8 (-) 9 is positive and -8 (-) 8 is negative, so the 10th term is greater than the 9th term. So, statements b, c, d, and e are all true. H.O.T. Focus on Higher Order Thinking. Justify Reasoning Suppose you are given a sequence with r < 0. What do you know about the signs of the terms of the sequence? Explain. Because r < 0, if the first term is negative, the second is positive. Then the third must be negative. The signs of the terms continue to alternate. 3. Critique Reasoning Miguel writes the following: 8, x, 8, x, He tells Alicia that he has written a geometric sequence and asks her to identify the value of x. Alicia says the value of x must be 8. Miguel says that Alicia is incorrect. Who is right? Explain. The value that Alicia gave is correct, but it is not the only correct value. x could also be -8. 8, -8, 8, -8,... is also a geometric sequence, so Miguel is correct. Module Lesson Lesson 1.1
12 Lesson Performance Task Multi-Step Gifford earns money by shoveling snow for the winter. He offers two payment plans: either pay $400 for the entire winter or pay $5 for the first week, $10 for the second week, $0 for the third week, and so on. Explain why each plan does or does not form a geometric sequence. Then determine the number of weeks after which the total cost of the second plan will exceed the total cost of the first plan. Each plan forms a geometric sequence. In the first plan, the geometric sequence is f (n) = 400. In the second plan, the geometric sequence is g (n) = 5 () n-1. Create a table of values with columns listing the week numbers, the weekly payments, and the total paid up to and including that week for the second plan. In week 6, the second plan s payment will be $160, and the total payments for weeks 1 6 will be $30, which is less than the $400 payment for the first plan. In week 7, the second plan s payment will be $30, and the total payments for weeks 1 7 will be $635, which is more than the $400 payment for the first plan. So, after 7 weeks, the total cost of the second plan will exceed the total cost of the first plan. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Guide students who need help in representing the geometric sequence 5, 10, 0, by giving them an explicit function. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students explain how they could use reasoning to check their answers for the approximate number of weeks when the plans cost about the same amount of money. For example, some students may say they wrote out the geometric sequence until they could see that $400 would occur between weeks 7 and 8. Image Credits: Trudy Wilkerson/Shutterstock Module Lesson 1 EXTENSION ACTIVITY Have students explain which of the following two options they would prefer. Have students explain their reasoning. Option 1: They receive 1 on the first day, on the second day, 4 on the third day, 8 on the fourth day, and so on, doubling each day, for a total of 0 days. Option : They receive $5000 on the first day and $0 after that. Students will find that with Option 1, on the 0th day, without including the amounts from the previous 19 days, the amount received would be $54.88, which is more than $5000. So, the total amount received in Option 1 would be much more than the total amount, $5000, received in Option. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Understanding Geometric Sequences 556
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