Integrated Math 1 Module 3 Arithmetic and Geometric Sequences

Transcription

1 1 Integrated Math 1 Module 3 Arithmetic and Geometric Sequences Adapted From The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius In partnership with the Utah State Office of Education 2012 Mathematics Vision Project MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution NonCommercial ShareAlike 3.0 Unported license

2 2 Prerequisite Concepts & Skills: Graph linear functions Evaluate expressions with exponents Identify patterns Module 3 Overview Summary of the Concepts & Skills in Module 3: Determine if a sequence is arithmetic or geometric Identify the common difference or common ratio in a sequence Find the next terms in arithmetic and geometric sequences Write recursive and explicit rules/formulas for arithmetic and geometric sequences Use function notation to evaluate functions Model sequences with a table of values and graph Find the arithmetic and geometric means of a given sequence Content Standards and Standards of Mathematical Practice Covered: Content Standards: F.BF.1, F.LE.1, F.LE.2, F.LE.5, A.REI.3 Standards of Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Materials: Graphing Utility Graph paper Teacher resource pages for each task (blank tables & graphs) Chart paper Pens Module 3 Vocabulary: Sequence Arithmetic sequence Geometric sequence Constant difference Constant ratio Function notation Recursive rule/formula Explicit rule/formula Model the data/sequence Arithmetic means Geometric means

3 3 Concepts Used In the Next Module: Use multiple representations of exponential functions Use multiple representations of linear functions State domain Compare growth rates of linear and exponential functions Solve linear and exponential functions Use simple and compound interest Three forms of linear functions: slope-intercept, standard, and point-slope

4 Module 3 Arithmetic and Geometric Sequences Representing arithmetic sequences with equations, tables, graphs, and story context (F.BF.1, F.LE.1, F.LE.2 F.LE.5) Warm Up: Creating a sentence describing a pattern/finding the next two terms Classroom Task: Growing Dots A Develop Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Representing geometric sequences with equations, tables, graphs, and story context (F.BF.1, F.LE.1, F.LE.2 F.LE.5) Warm Up: Use a graphing utility to confirm graphs for 3.1 "GO are correct Classroom Task: Growing, Growing Dots A Develop Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Arithmetic sequences: Constant difference between consecutive terms and Geometric Sequences: Constant ratio between consecutive terms (F.BF.1, F.LE.1, F.LE.2, F.LE.5) Warm Up: Scott s Workout A Solidify Understanding Task Classroom Task: Don t Break the Chain A Solidify Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Arithmetic Sequences: Increasing and decreasing at a constant rate (F.BF.1, F.LE.1, F.LE.2 F.LE.5) Warm Up: Finding the Constant Difference Classroom Task: Something to Chew On A Solidify Understanding Task Set, Go Homework: Arithmetic and Geometric Sequences Comparing rates of growth in arithmetic and geometric sequences (F.BF.1, F.LE.1, F.LE.2 F.LE.5) Classroom Task: Chew On This A Solidify Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Recursive and explicit equations for arithmetic and geometric sequences (F.BF.1, F.LE.1, F.LE.2) Warm Up: Multiple Representations of a Sequence Classroom Task: What Comes Next? What Comes Later? A Solidify Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Using rate of change to find missing terms in an arithmetic sequence and Using a constant ratio to find missing terms in a geometric sequence (A.REI.3) Warm Up: Comparing arithmetic and geometric sequences Classroom Task: What Does It Mean? A Solidify Understanding Task Geometric Meanies A Solidify and Practice Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Developing fluency with geometric and arithmetic sequences (F.LE.2) Warm Up: Explicit Equations of Geometric Equations Classroom Task: I Know... What Do You Know? A Practice Understanding Task Ready, Set, Go Homework: Arithmetic and Geometric Sequences Module 3 Review Classroom Task: Carousel Activity Homework: Arithmetic and Geometric Sequences 3.9

5 5 3.1 Warm Up Identifying a Pattern Write a sentence to describe the pattern in each problem. Then find the next two terms in the list. 1. Pattern Description: 2. Pattern Description: Next Two Terms: 3. Pattern Description: Next Two Terms: 4. Pattern Description: Next Two Terms: Next Two Terms: 5. Create your own sequence. Swap sequences with a partner. Describe the pattern and find the next two terms.

6 Growing Dots A Develop Understanding Task Solve the problems by your preferred method. Be sure to show how your solution relates to the picture and how you arrived at your solution. 1. Describe the pattern that you see in the sequence of figures above. 2. Assuming the sequence continues in the same way, how many dots are there in 3 minutes? 3. How many dots are there at 100 minutes? 4. How many dots are there at t minutes?

7 7 Name: Arithmetic and Geometric Sequences 3.1 Ready, Set, Go! Ready Topic: Exponents, substitution, and function notation Find each value For each of the following, find, and Complete each table Term 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Value Term 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Value Term 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Value 9 81 Term 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Value Term 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th Value 5 12

8 8 Set Topic: Completing a table Fill in the table. Then write a sentence explaining how you figured out the values to put in each cell. Explain how to figure out what will be in cell # You run a business making birdhouses. You spend $600 to start your business, and it costs you $5.00 to make each birdhouse. # of birdhouses Total cost to build Explanation: 14. You borrow $500 from a relative, and you agree to pay back the debt at a rate of $15 per month. # of months Amount of money owed Explanation: 15. You earn $10 per week. # of weeks Amount of money earned Explanation: 16. You are saving for a bike and can save $10 per week. You have $25 already saved. # of weeks Amount of money saved Explanation:

9 9 Go Topic: Good viewing window When sketching a graph of a function, it is important that we see key points. For linear functions, we want a window that shows important information related to the story. Often, this means including both the x- and y- intercepts Find an appropriate graphing window for each of the following linear functions. Fill in the blanks showing the lower and upper values and include the scale for each axis

10 Warm Up Graphing Windows For #1-4, use graphing utility to confirm that the graphs you created in the 3.1 Go homework from last night were correct. 1. a. Does the graph you created last night look like the one on your graphing utility? 2. a. Does the graph you created last night look like the one on your graphing utility? b. Explain why your graph did or did not look like the one on your graphing utility. b. Explain why your graph did or did not look like the one on your graphing utility. c. Are the x- and y- intercepts visible on both graphs? c. Are the x- and y- intercepts visible on both graphs? 3. a. Does the graph you created last night look like the one on your graphing utility? 4. a. Does the graph you created last night look like the one on your graphing utility? b. Explain why your graph did or did not look like the one on your graphing utility. b. Explain why your graph did or did not look like the one on your graphing utility. c. Are the x- and y- intercepts visible on both graphs? c. Are the x- and y- intercepts visible on both graphs? 5. Use the arithmetic sequence below to answer the following questions: a. What is the common difference? b. What is the recursive rule/formula? c. What is the explicit rule/formula?

11 Growing, Growing Dot A Develop Understanding Task At the beginning At one minute At two minutes At three minutes At four minutes 1. Describe and label the pattern of change you see in the above sequence of figures. 2. Assuming the sequence continues in the same way, how many dots are there at 5 minutes? 3. Write a recursive formula to describe how many dots there will be after t minutes. 4. Write an explicit formula to describe how many dots there will be after t minutes.

12 12 Name: Arithmetic and Geometric Sequences 3.2 Ready, Set, Go! Ready Topic: Finding values for a pattern 1. Bob Cooper was born in By 1930 he had 3 sons, all with the Cooper last name. By 1960 each of Bob s 3 boys had exactly 3 sons of their own. By the end of each 30 year time period, the pattern of each Cooper boy having exactly 3 sons of their own continued. How many Cooper sons were born in the 30 year period between 1960 and 1990? 2. Create a diagram that would show this pattern. 3. Predict how many Cooper sons will be born between 1990 and 2020, if the pattern continues. 4. Try to write an equation that would help you predict the number of Cooper sons that would be born between 2020 and If you can t find the equation, explain it in words. Set Topic: Evaluating equations Evaluate the following equations when { }. Organize your inputs and outputs into a table of values for each equation. Let x be the input and y be the output x y x y x y x y

13 13 Go Topic: Solve equations Solve the following equations for the unknown variable. Check your answer ( ) ( )

14 Warm Up: Scott s Workout A Solidify Understanding Task Scott has decided to add push-ups to his daily exercise routine. He is keeping track of the number of push-ups he completes each day in the bar graph below, with day one showing he completed three push-ups. After four days, Scott is certain he can continue this pattern of increasing the number of push-ups he completes each day. 1. How many push-ups will Scott do on day 10? 2. How many push-ups will Scott do on day n? 3. Model the number of push-ups Scott will complete on any given day. Include both explicit and recursive equations. (Days) # of Push Ups Recursive equation: Explicit equation: 4. Aly is also including push-ups in her workout and says she does more push-ups than Scott because she does fifteen push-ups every day. Is she correct? Explain.

15 Don t Break the Chain A Solidify Understanding Task Maybe you ve received an like this before: Hi! My name is Bill Weights, founder of Super Scooper Ice Cream. I am offering you a gift certificate for our signature Super Bowl (a $4.95 value) if you forward this letter to 10 people. When you have finished sending this letter to 10 people, a screen will come up. It will be your Super Bowl gift certificate. Print that screen out and bring it to your local Super Scooper Ice Cream store. The server will bring you the most wonderful ice cream creation in the world a Super Bowl with three yummy ice cream flavors and three toppings! This is a sales promotion to get our name out to young people around the country. We believe this project can be a success, but only with your help. Thank you for your support. Sincerely, Bill Weights Founder of Super Scooper Ice Cream These chain s rely on each person that receives the to forward it on. Have you ever wondered how many people might receive the if the chain remains unbroken? To figure this out, assume that it takes a day for the to be opened, forwarded, and then received by the next person. On day 1, Bill Weights starts by sending the out to his 8 closest friends. They each forward it to 10 people so that on day 2, it is received by 80 people. The chain continues unbroken. 1. How many people will receive the on day 7? 2. How many people with receive the on day n? Explain your answer with as many representations as possible. 3. If Bill gives away a Super Bowl that costs $4.95 to every person that receives the during the first week, how much will he have spent?

16 16 Name: Arithmetic and Geometric Sequences 3.3 Ready, Set, Go! Ready Topic: Write the equation of a line given two points. Graph each pair of points, draw a line that goes through both points, and write an equation of that line. 1. and 2. and Equation: Equation: 3. and 4. and Equation: Equation: 5. Write the equation of the line that passes through the points and without the help of a graph.

17 17 Set Topic: Recursive and explicit functions of arithmetic sequences Below you are given various types of information. Write the recursive and explicit functions for each arithmetic sequence. Finally, graph each sequence, making sure you clearly label your axes Time (Days) # of cells Recursive: Explicit: 8. Claire has $300 in an account. She decides she is going to take out $25 each month. Recursive: Explicit: 9. Each day Tania decides to do something nice for 2 strangers. Write recursive and explicit equations that represent the number of strangers Tania that does something nice for each day (not total number of strangers). Recursive: Explicit: Recursive: Explicit:

18 Recursive: Explicit: Go Topic: Recursive and explicit functions of geometric sequences Below you are given various types of information. Write the recursive and explicit functions for each geometric sequence. Finally, graph each sequence, making sure you clearly label your axes Time # of (Days) cells Recursive: Explicit: Recursive: Explicit:

19 Claire has $300 in an account. She decides she is going to take out half of what s left in there at the end of each month. 14. Tania creates a chain letter and sends it to four friends. Each day each friend is then instructed to send it to four friends and so forth. Recursive: Explicit: Recursive: Explicit: 15. Recursive: Explicit:

20 Warm Up Finding the constant difference Find the missing terms for each arithmetic sequence and state the constant difference. 1. 5, 11,, 23, 29, 2. 7, 3,,,,, Constant Difference = Constant Difference = 3. 8,,, 47, 60, 4. 0,,, 2,, Constant Difference = Constant Difference = 5. 5,,,, 25, 6. 3,,,,, Constant Difference = Constant Difference =

21 Something to Chew On A Solidify Understanding Task The Food-Mart grocery store has a candy machine like the one pictured here. Each time a child inserts a quarter, 7 candies come out of the machine. The machine holds 15 pounds of candy. Each pound of candy contains about 180 individual candies. 1. Represent the number of candies in the machine for any given number of customers. 2. About how many customers will there be before the machine is empty? 3. Represent the amount of money in the machine for any given number of customers. 4. To avoid theft, the store owners don t want to let too much money collect in the machine, so they take all the money out when they think the machine has about $25 in it. The tricky part is that the store owners can t tell how much money is actually in the machine without opening it up, so they choose when to remove the money by judging how many candies are left in the machine. About how full should the machine look when they take the money out? How do you know?

22 5. Create a different context that illustrates an arithmetic sequence (Do not use the same context as the task). When you have come up with a context, create two follow up questions that can be asked. Swap your problem with another group and solve their problem. 22 Follow up question #1: Table: Graph: Equation Follow up question #2:

23 23 Name: Arithmetic and Geometric Sequences 3.4 Set, Go! Set Topic: Determine recursive equations Two consecutive terms in an arithmetic sequence are given. Find the constant difference and the recursive equation. 1. If Recursive Function: 2. If Recursive Function: 3. If Go Topic: Evaluate using function notation Recursive Function: Find each value. 4.. Find. 5.. Find. 6.. Find. 7.. Find and. 8. Find and.

24 Chew on This A Solidify Understanding Task Mr. and Mrs. Gloop want their son, Augustus, to do his homework every day. Augustus loves to eat candy, so his parents have decided to motivate him to do his homework by giving him candies for each day that the homework is complete. Mr. Gloop says that on the first day that Augustus turns in his homework, he will give him 10 candies. On the second day he promises to give 20 candies, on the third day he will give 30 candies, and so on. 1. Write both a recursive and an explicit formula that shows the number of candies that Augustus earns on any given day with his father s plan. Recursive Formula: Explicit Formula: 2. Use a formula to find how many candies Augustus will have on day 30 in this plan. Augustus looks in the mirror and decides that he is gaining weight. He is afraid that all that candy will just make it worse, so he tells his parents that it would be ok if they just give him 1 candy on the first day, 2 on the second day, continuing to double the amount each day as he completes his homework. Mr. and Mrs. Gloop likes Augustus plan and agrees to it. 3. Model the amount of candy that Augustus would get each day he reaches his goals with the new plan. # of Days # of Candies 4. Use your model to predict the number of candies that Augustus would earn on the 30 th day with this plan.

25 25 5. Write both a recursive and an explicit formula that shows the number of candies that Augustus earns on any given day with this plan. Recursive Formula: Explicit Formula: Augustus is generally selfish and somewhat unpopular at school. He decides that he could improve his image by sharing his candy with everyone at school. When he has a pile of 100,000 candies, he generously plans to give away 60% of the candies that are in the pile each day. Although Augustus may be earning more candies for doing his homework, he is only giving away candies from the pile that started with 100,000. (He s not that generous.) 6. Model the amount of candy that would be left in the pile each day. # of Days # of Candies 7. How many pieces of candy will be left on day 8? 8. When would the candy be gone?

26 26 Name: Arithmetic and Geometric Sequences 3.5 Ready, Set, Go! Ready Topic: Arithmetic and geometric sequences Find the missing values for each arithmetic or geometric sequence. Then say if the sequence has a constant difference or a constant ratio, and say what the constant difference/rate is Constant difference or a constant ratio? Constant difference or a constant ratio? The constant difference/ratio is The constant difference/ratio is Constant difference or a constant ratio? Constant difference or a constant ratio? The constant difference/ratio is The constant difference/ratio is Set Topic: Recursive and explicit equations Determine whether each situation represents an arithmetic or geometric sequence and then find the recursive and explicit equation for each Arithmetic or Geometric? Arithmetic or Geometric? Recursive: Recursive: Explicit: Explicit:

27 27 7. Time (days) Number of Dots Time (days) Number of Cells Arithmetic or Geometric? Arithmetic or Geometric? Recursive: Recursive: Explicit: Explicit: 9. Scott decides to add running to his exercise routine and runs a total of one mile his first week. He plans to double the number of miles he runs each week. 10. Michelle likes chocolate so much that she eats it every day and it always 3 more pieces than the previous day. She ate 3 pieces on day 1. Arithmetic or Geometric? Arithmetic or Geometric? Recursive: Recursive: Explicit: 11. Vanessa has $60 to spend on rides at the State Fair. Each ride cost $4. Explicit: 12. Cami invested $6,000 dollars into an account that earns 10% interest each year. Arithmetic or Geometric? Arithmetic or Geometric? Recursive: Recursive: Explicit: Explicit:

28 28 Go Topic: Solving systems of linear equations Solve the system of equations. 13. { 14. { 15. { 16. {

29 Warm Up Explicit and Recursive Rules For each of the following tables, Identify if the function is arithmetic, geometric or neither Describe how to find the next term in the sequence, Write a recursive rule for the function, Describe how the features identified in the recursive rule can be used to find the value of the n th term, Write an explicit rule for the function, and Determine if the graph would be a straight line or curved and WHY. ***NOTE: If you recognize that the function is NEITHER, you do not need to find the recursive rule. Example: x y ? n? Arithmetic, geometric, or neither? To find the next term: Recursive rule: To find the n th term: Explicit rule: Would the graph be a straight line or curved? Your Turn x y ? n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion.

30 What Comes Next? What Comes Later? A Practice Understanding Task Function A x y 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: n? 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion. Function B x y n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion.

31 31 Function C X y ? n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion. Function D x y ? n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion.

32 32 Function E x y 1. Arithmetic, geometric, or neither? To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion. n? Function F x y n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion.

33 33 Function G x 1 y n? 1. Arithmetic, geometric, or neither? 2. To find the next term: 3. Recursive rule: 4. To find the n th term: 5. Explicit rule: 6. Would the graph of this data be a straight line or a curve? Explain how you reached this conclusion. How do you use Common ratio or difference First term Arithmetic Sequence Recursive Formula Arithmetic Sequence Explicit Formula Geometric Sequence Recursive Formula Geometric Sequence Explicit Formula

34 34 Name: Arithmetic and Geometric Sequences 3.6 Ready, Set, Go! Ready Topic: Constant Ratios Find the constant ratio for each geometric sequence Set Topic: Recursive and explicit equations Fill in the blanks for each table and then write the recursive and explicit equation for each sequence. 5. Table Recursive: Explicit: 6. Table 2 7. Table 3 8. Table 4 x y x y x y Recursive: Recursive: Recursive: Explicit: Explicit: Explicit:

35 35 Go Topic: Graphing linear equations and labeling your axes. Graph the following linear equations. Label your axes

36 Warm Up Comparing Arithmetic and Geometric Sequences 1. How are arithmetic and geometric sequences similar? 2. How are they different? 3. Give an example of each. Arithmetic: Geometric:

37 Part 1: What Does It Mean? A Solidify & Practice Understanding Task Each of the tables below represents an arithmetic sequence. Find the missing terms in the sequence, showing your method. Table Table Table Describe your method for finding the missing terms. Will the method always work? How do you know?

38 38 Here are a few more arithmetic sequences with missing terms. Complete each table, either using the method you developed previously or by finding a new method. Table Table Table The missing terms in an arithmetic sequence are called arithmetic means. For example, in the problem above, you might say, Find the 6 arithmetic means between and 5. Describe a method that will work to find arithmetic means and explain why this method works.

39 Part 2: Geometric Meanies A Solidify & Practice Understanding Task Each of the tables below represents a geometric sequence. Find the missing terms in the sequence, showing your method. Table Is the missing term that you identified the only answer? Why or why not? Table Are the missing terms that you identified the only answers? Why or why not? Table Are the missing terms that you identified the only answers? Why or why not?

40 40 Table Are the missing terms that you identified the only answers? Why or why not? a. Describe your method for finding the geometric means. b. How can you tell if there will be more than one solution for the geometric means?

41 41 Name: Arithmetic and Geometric Sequences 3.7 Ready, Set, Go! Ready Topic: Arithmetic and geometric sequences For each set of sequences, find the first five terms. Compare arithmetic sequences and geometric sequences. Which grows faster? When? 1. Arithmetic sequence: common difference, Geometric sequence: common ratio, Arithmetic: Geometric: Which value do you think will be more, or? Why? 2. Arithmetic sequence: common difference, Geometric sequence: common ratio, Arithmetic: Geometric: Which value do you think will be more, or? Why?

42 42 Set Topic: Arithmetic sequences Each of the tables below represents an arithmetic sequence. Find the missing terms in the sequence, showing your method. 3. Table Table 2 5. Table 3 6. Table Topic: Geometric sequences Each of the tables below represents a geometric sequence. Find the missing terms in the sequence, showing your method. 7. Table Table 2 9. Table Table

43 43 Go Topic: Sequences Determine the recursive and explicit equations for each (if the sequence is not arithmetic or geometric, try your best). 11. This sequence is: Arithmetic, Geometric, Neither Recursive Equation: Explicit Equation: 12. This sequence is: Arithmetic, Geometric, Neither Recursive Equation: Explicit Equation: 13. This sequence is: Arithmetic, Geometric, Neither Recursive Equation: Explicit Equation: 14. (The percentage of tiles shaded black) This sequence is: Arithmetic, Geometric, Neither Recursive Equation: Explicit Equation: 15. This sequence is: Arithmetic, Geometric, Neither Recursive Equation: Explicit Equation:

44 Warm Up Explicit Equations of Geometric Equations Given the following information, determine the explicit equation for each geometric sequence. 1. common ratio, Which geometric sequence above has the greatest value at?

45 I Know... What Do You Know? A Practice Understanding Task In each of the problems below I share some of the information that I know about a sequence. Your job is to find a minimum of 4 of the following things for each sequence: a table, a graph, an explicit equation, a recursive equation, the constant ratio or constant difference between consecutive terms, any terms that are missing, the type of sequence, and a story context. 1. I know that: the recursive formula for the sequence is. What do you know? 2. I know that: the first 5 terms of the sequence are What do you know? 3. I know that: the explicit formula for the sequence is. What do you know?

46 46 4. I know that: the first 4 terms of the sequence are What do you know? 5. I know that: the sequence is arithmetic and and. What do you know? 6. I know that: the sequence is a model for the perimeter of the following figures: What do you know?

47 7. I know that: it is a sequence where and the constant ratio between terms is. What do you know? I know that: the sequence models the value of a car that originally cost $26,500, but loses 10% of its value each year. What do you know? 9. I know that: the first term of the sequence is, and the fifth term is. What do you know? 10. I know that: a graph of the sequence is: What do you know?

48 48 Name: Arithmetic and Geometric Sequences 3.8 Ready, Set, Go! Ready Topic: Comparing linear equations and arithmetic sequences 1. Describe similarities and differences between linear equations and arithmetic sequences. Similarities Differences Set Topic: representations of arithmetic sequences Use the given information to complete the other representations for each arithmetic sequence. 2. Recursive Equation: Graph: Explicit Equation: Table: Days Cost Create a Context:

49 49 3. Recursive Equation: Graph: Explicit Equation: Table: Create a Context: 4. Recursive Equation: Graph: Explicit Equation: Table: Create a Context:

50 50 5. Recursive Equation: Graph: Explicit Equation: Table: Create a Context: Janet wants to know how many seats are in each row of the theater. Jamal lets her know that each row has 2 seats more than the row in front of it. The first row has 14 seats. Go Topic: Writing explicit equations Given the recursive equation for each arithmetic sequence, write the explicit equation

51 51 Name: Arithmetic and Geometric Sequences 3.9 Use the given information to state as much as possible about each sequence. Your answer should include: type of sequence, the common difference or common ration, a table of at least 5 terms, a graph, the recursive rule, and the explicit rule. 1. Type: Common difference/ratio: Recursive rule:, Explicit rule: 2. Type: Common difference/ratio: Recursive rule: Explicit rule:

52 52 3. Type: 1 3 Common difference/ratio: 2 Recursive rule: Explicit rule: 4. Type: 1 40 Common Ratio = Recursive rule: Explicit rule: 5. Explain how you tell if a sequence is arithmetic and if a sequence is geometric.

53 53 For #6-8, determine if each sequence is arithmetic or geometric. Find the values of the next two terms. Then write the explicit and recursive formulas for each sequence. 6. Arithmetic or Geometric Next 2 terms: Recursive Formula: Explicit Formula: 7. Arithmetic or Geometric Next 2 terms: Recursive Formula: Explicit Formula: 8. Arithmetic or Geometric Next 2 terms: Recursive Formula: Explicit Formula: 9. Find the missing terms of the arithmetic sequence below. Be sure to show all work