Chapter 4 STUDY GUIDE Sequences Name: __________________________ Hour: __________________________ 1. Consider the sequence shown. a. Draw the next two figures of the pattern. b. Describe the pattern. c. Write a numeric sequence to represent the first 5 figures. d. 2. Consider the sequence shown. a. Draw the next two figures of the pattern. b. Write a numeric sequence to represent the first 5 figures. c. Describe the sequence using the terms arithmetic sequence, geometric sequence, common difference, and/or common ratio as they apply. d. Write the explicit formula for this sequence. Chapter 4 STUDY GUIDE Name: __________________________ Sequences Hour: __________________________ 3. Identify each sequence as arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, determine the common difference or common ratio. a. 1, 3, 5, 7, 9, 11 Sequence Type: __________________________ b. -2, 6, -18, 54, -162 Sequence Type: __________________________ c. Common Difference/ Ratio: ________________ 1, 4, 9, 16, 25 Sequence Type: __________________________ d. Common Difference/ Ratio: ________________ 1 1 1 Common Difference/ Ratio: ________________ 1 1, 3 , 9 , 27 , 81 Sequence Type: __________________________ Common Difference/ Ratio: ________________ 4. For each sequence, determine whether the sequence is arithmetic or geometric and write an explicit formula. Using your formula, determine the 12th term in the sequence. a. 4, 8, 16, 32, 64, 128 Arithmetic or Geometric Explicit Formula: _______________________ 12th term: _____________________________ b. 3 5 7 1, 2 , 2, 2 , 3, 2 , 4 Arithmetic or Geometric Explicit Formula: _______________________ 12th term: _____________________________ c. -1, -0.25, 0.5, 1.25, 2 Arithmetic or Geometric Explicit Formula: _______________________ 12th term: _____________________________ Chapter 4 STUDY GUIDE Name: __________________________ Sequences Hour: __________________________ 5. For each sequence, use the recursive formula to determine the next three terms in the sequence. Show your work. a. Let 𝑎1 = 12 and 𝑎𝑛 = 𝑎𝑛−1 + 4 𝑎2 = 𝑎3 = 𝑎4 = b. 1 1 Let 𝑎1 = 5 and 𝑎𝑛 = 𝑎𝑛−1 (2) 𝑎2 = 𝑎3 = 𝑎4 = 6. Rewrite each explicit formula in simplest form. a. 𝑎𝑛 = 4 + 3(𝑛 − 1) b. 𝑎𝑛 = 2 − 5(𝑛 − 1) 6a. ________________________________ 6b. _____________________________ 7. Marylins’s Bakery made 15 cakes on Monday, 22 cakes on Tuesday, and 29 cakes on Wednesday. If this pattern continues, how many cakes will Marylin’s Bakery make on Friday? 7. ________________ 8. The Johnsons are draining their family swimming pool. After one-half hour, there are 8300 gallons of water in the pool. After one hour, there are 7900 gallons of water in the pool. After one and one-half hours, there are 7500 gallons of water in the pool. If this pattern continues, how much water will be in the pool after 3 hours? 8. ________________ Chapter 4 STUDY GUIDE Name: __________________________ Sequences Hour: __________________________ 9. Jesse makes two text messages to his friends to tell them school is cancelled because of storm flooding. Each of those friends makes two text messages to tell their friends the same news. Each of those friends makes two text messages to tell their friends the same news, and so on. a. Write a numeric sequence to represent the number of text messages made in each of the first 5 sets of text messages. b. Is this an arithmetic or geometric sequence? 9a. ____________________________________ 9b. ____________________________________ 10. Write the first 4 terms of each sequence. a. an arithmetic sequence with a common difference of 6 and a first term of 10 10a. ________________________________________ b. a geometric sequence with a common ratio of 1 2 and a first term of 48 10b. ________________________________________ c. an arithmetic sequence with a common difference of 0.25 and a first term of 9 10c. ________________________________________ d. a geometric sequence with a common ratio of 4 and a first term of 2 10d. ________________________________________ 11. Determine the 50th term in the sequence. 𝑎𝑛 = −10 + 4(𝑛 − 1). 11. ___________ 12. Determine the 7th term in the sequence. 𝑎𝑛 = 1 3 (3)𝑛−1 . 12. ___________ Chapter 4 STUDY GUIDE Sequences Name: __________________________ Hour: __________________________ 13. Tell whether each graph represents an arithmetic sequence or a geometric sequence. Explain your reasoning. a. b. __________________________________ __________________________________ __________________________________ __________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ____ 14. Which statement describes the pattern shown? a. b. c. d. Each figure has 3 fewer squares than the one before it. Each figure has 6 fewer squares than the one before it. Each figure has 3 more squares than the one before it. Each figure has 6 more squares than the one before it. ____ 15. George has agreed to donate $200 to Sun Valley High School for its library. In addition, he will donate $3 for every book a student at Sun Valley High School reads during the summer. The sequence shown represents the possible amounts that George will be donating for the summer. 200, 203, 206, 209, 212, 215… Which explicit formula represents this problem situation? a. b. c. d. 𝑎𝑛 𝑎𝑛 𝑎𝑛 𝑎𝑛 = 3 + 200(𝑛 − 1) = 200 + 3(𝑛 − 1) = 200 + (𝑛 − 3) = 250 + 3𝑛 Chapter 4 STUDY GUIDE Sequences Name: __________________________ Hour: __________________________ ____ 16. Which sequence has a common ratio of 4? a. b. c. d. 9, 5, 1, 3, -7 1, 4, 16, 64, 256 9, 13, 17, 21, 25 1, 4, 16, 64, 256 ____ 17. Which represents the explicit formula for the arithmetic sequence 𝑎𝑛 = 12 + 4(𝑛 − 1) in simplest form? a. b. c. d. 𝑓(𝑛) = 𝑛 + 8 𝑓(𝑛) = 4𝑛 + 12 𝑓(𝑛) = 𝑛 + 16 𝑓(𝑛) = 4𝑛 + 8 ____ 18. Which sequence has a common difference of 1.75? a. b. c. d. 0, 1.75, 3.5, 5.25, -7 0, 1.75, 3.5, 5.25, 7 0, 0.75, 1.5, 2.25, 3 0, 0.75, 1.5, 2.25, -3 ____ 19. Determine if the sequence is arithmetic or geometric. Then identify the next term in the sequence. 0.25, 1, 4, 16,… a. b. c. d. arithmetic; 32 arithmetic; 64 geometric; 32 geometric; 64 Explicit Formulas 𝒂𝒏 = 𝒂𝟏 + 𝒅(𝒏 − 𝟏) 𝒂𝒏 = 𝒂𝟏 (𝒓)𝒏−𝟏 Recursive Formulas 𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒅 𝒂𝒏 = 𝒂𝒏−𝟏 (𝒓) Chapter 4 STUDY GUIDE Sequences Name: __________________________ Hour: __________________________