11.2 Arithmetic and Geometric Sequences, Exams of Calculus

Download 11.2 Arithmetic and Geometric Sequences and more Exams Calculus in PDF only on Docsity! 640 CHAPTER 11 Sequences, Series, and the Binomial Theorem 53. 12, -72 and 1-3, -32 54. 110, -142 and 15, -112 CONCEPT EXTENSIONS Find the first five terms of each sequence. Round each term after the first to four decimal places. 55. an = 1 2n 56. 2n 2n + 1 57. an = a1 + 1 n bn 58. an = a1 + 0.05 n bn 46. A Fibonacci sequence is a special type of sequence in which the first two terms are 1, and each term thereafter is the sum of the two previous terms: 1, 1, 2, 3, 5, 8, etc. The formula for the nth Fibonacci term is an = 1 15 c a1 + 15 2 bn - a1 - 15 2 bn d . Verify that the first two terms of the Fibonacci sequence are each 1. REVIEW AND PREVIEW Sketch the graph of each quadratic function. See Section 8.5. 47. f 1x2 = 1x - 122 + 3 48. f 1x2 = 1x - 222 + 1 49. f 1x2 = 21x + 422 + 2 50. f 1x2 = 31x - 322 + 4 Find the distance between each pair of points. See Section 7.3. 51. 1-4, -12 and 1-7, -32 52. 1-2, -12 and 1-1, 52 OBJECTIVE 11.2 Arithmetic and Geometric Sequences OBJECTIVES 1 Identify Arithmetic Sequences and Their Common Differences. 2 Identify Geometric Sequences and Their Common Ratios. 1 Identifying Arithmetic Sequences Find the first four terms of the sequence whose general term is an = 5 + 1n - 123. a1 = 5 + 11 - 123 = 5 Replace n with 1. a2 = 5 + 12 - 123 = 8 Replace n with 2. a3 = 5 + 13 - 123 = 11 Replace n with 3. a4 = 5 + 14 - 123 = 14 Replace n with 4. The first four terms are 5, 8, 11, and 14. Notice that the difference of any two successive terms is 3. 8 - 5 = 3 11 - 8 = 3 14 - 11 = 3 f an - an-1 = 3 c c nth previous term term Because the difference of any two successive terms is a constant, we call the sequence an arithmetic sequence, or an arithmetic progression. The constant difference d in successive terms is called the common difference. In this example, d is 3. Arithmetic Sequence and Common Difference An arithmetic sequence is a sequence in which each term (after the first) differs from the preceding term by a constant amount d. The constant d is called the common difference of the sequence. The sequence 2, 6, 10, 14, 18, cis an arithmetic sequence. Its common difference is 4. Given the first term a1 and the common difference d of an arithmetic sequence, we can find any term of the sequence. Section 11.2 Arithmetic and Geometric Sequences 641 PRACTICE EXAMPLE 1 Write the first five terms of the arithmetic sequence whose first term is 7 and whose common difference is 2. Solution a1 = 7 a2 = 7 + 2 = 9 a3 = 9 + 2 = 11 a4 = 11 + 2 = 13 a5 = 13 + 2 = 15 The first five terms are 7, 9, 11, 13, 15. 1 Write the first five terms of the arithmetic sequence whose first term is 4 and whose common difference is 5. Notice the general pattern of the terms in Example 1. a1 = 7 a2 = 7 + 2 = 9 or a2 = a1 + d a3 = 9 + 2 = 11 or a3 = a2 + d = 1a1 + d2 + d = a1 + 2d a4 = 11 + 2 = 13 or a4 = a3 + d = 1a1 + 2d2 + d = a1 + 3d a5 = 13 + 2 = 15 or a5 = a4 + d = 1a1 + 3d2 + d = a1 + 4d T 1subscript - 12 is multiplier c The pattern on the right suggests that the general term an of an arithmetic se- quence is given by an = a1 + 1n - 12d General Term of an Arithmetic Sequence The general term an of an arithmetic sequence is given by an = a1 + 1n - 12d where a1 is the first term and d is the common difference. EXAMPLE 2 Consider the arithmetic sequence whose first term is 3 and whose common difference is -5. a. Write an expression for the general term an. b. Find the twentieth term of this sequence. Solution a. Since this is an arithmetic sequence, the general term an is given by an = a1 + 1n - 12d. Here, a1 = 3 and d = -5, so an = 3 + 1n - 121-52 Let a1 = 3 and d = -5. = 3 - 5n + 5 Multiply. = 8 - 5n Simplify. b. an = 8 - 5n a20 = 8 - 5 # 20 Let n = 20. = 8 - 100 = -92 644 CHAPTER 11 Sequences, Series, and the Binomial Theorem PRACTICE PRACTICE EXAMPLE 8 Find the fifth term of the geometric sequence whose first three terms are 2, -6, and 18. Solution Since the sequence is geometric and a1 = 2, the fifth term must be a1 r5-1, or 2r4. We know that r is the common ratio of terms, so r must be -6 2 , or -3. Thus, a5 = 2r4 a5 = 21-324 = 162 8 Find the seventh term of the geometric sequence whose first three terms are -3, 6, and -12. PRACTICE EXAMPLE 7 Find the eighth term of the geometric sequence whose first term is 12 and whose common ratio is 1 2 . Solution Since this is a geometric sequence, the general term an is given by an = a1 rn-1 Here a1 = 12 and r = 1 2 , so an = 12a 1 2 bn-1 . Evaluate an for n = 8. a8 = 12a 1 2 b8-1 = 12a 1 2 b7 = 12a 1 128 b = 3 32 7 Find the seventh term of the geometric sequence whose first term is 64 and whose common ratio is 1 4 . Notice the general pattern of the terms in Example 6. a1 = 7 a2 = 7122 = 14 or a2 = a11r2 a3 = 14122 = 28 or a3 = a21r2 = 1a1 # r2 # r = a1 r2 a4 = 28122 = 56 or a4 = a31r2 = 1a1 # r22 # r = a1 r3 a5 = 56122 = 112 or a5 = a41r2 = 1a1 # r32 # r = a1 r4 d T 1subscript - 12 is power The pattern on the right above suggests that the general term of a geometric sequence is given by an = a1 rn-1. General Term of a Geometric Sequence The general term an of a geometric sequence is given by an = a1 rn-1 where a1 is the first term and r is the common ratio. 6 Write the first four terms of a geometric sequence whose first term is 8 and whose common ratio is -3 Section 11.2 Arithmetic and Geometric Sequences 645 PRACTICE EXAMPLE 9 If the second term of a geometric sequence is 5 4 and the third term is 5 16 , find the first term and the common ratio. Solution Notice that 5 16 , 5 4 = 1 4 , so r = 1 4 . Then a2 = a1a 1 4 b2-1 5 4 = a1a 1 4 b1 , or a1 = 5 Replace a2 with 5 4 . The first term is 5. 9 If the second term of a geometric sequence is 9 2 and the third term is 27 4 , find the first term and the common ratio. 1. A(n) sequence is one in which each term (after the first) is obtained by multiplying the preceding term by a constant r. The constant r is called the common . 2. A(n) sequence is one in which each term (after the first) differs from the preceding term by a constant amount d. The constant d is called the common . 3. The general term of an arithmetic sequence is an = a1 + 1n - 12d where a1 is the term and d is the common . 4. The general term of a geometric sequence is an = a1r n-1 where a1 is the term and r is the common . Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Some choices may be used more than once and some not at all. first arithmetic difference last geometric ratio PRACTICE EXAMPLE 10 Predicting Population of a Bacterial Culture The population size of a bacterial culture growing under controlled conditions is doubling each day. Predict how large the culture will be at the beginning of day 7 if it measures 10 units at the beginning of day 1. Solution Since the culture doubles in size each day, the population sizes are modeled by a geometric sequence. Here a1 = 10 and r = 2. Thus, an = a1 rn-1 = 10122n-1 and a7 = 101227-1 = 640 The bacterial culture should measure 640 units at the beginning of day 7. 10 After applying a test antibiotic, the population of a bacterial culture is reduced by one-half every day. Predict how large the culture will be at the start of day 7 if it measures 4800 units at the beginning of day 1. 646 CHAPTER 11 Sequences, Series, and the Binomial Theorem 1. a1 = 4; d = 2 2. a1 = 3; d = 10 3. a1 = 6; d = -2 4. a1 = -20; d = 3 5. a1 = 1; r = 3 6. a1 = -2; r = 2 7. a1 = 48; r = 1 2 8. a1 = 1; r = 1 3 Find the indicated term of each sequence. See Examples 2 and 7. 9. The eighth term of the arithmetic sequence whose first term is 12 and whose common difference is 3 10. The twelfth term of the arithmetic sequence whose first term is 32 and whose common difference is -4 11. The fourth term of the geometric sequence whose first term is 7 and whose common ratio is -5 12. The fifth term of the geometric sequence whose first term is 3 and whose common ratio is 3 13. The fifteenth term of the arithmetic sequence whose first term is -4 and whose common difference is -4 14. The sixth term of the geometric sequence whose first term is 5 and whose common ratio is -4 Find the indicated term of each sequence. See Examples 3 and 8. 15. The ninth term of the arithmetic sequence 0, 12, 24, c 11.2 Exercise Set Write the first five terms of the arithmetic or geometric sequence, whose first term, a1, and common difference, d, or common ratio, r, are given. See Examples 1 and 6. OBJECTIVE 1 OBJECTIVE 2 Watch the section lecture video and answer the following questions. 5. From the lecture before Example 1, what makes a sequence an arith- metic sequence? 6. From the lecture before Example 3, what’s the difference between an arithmetic and a geometric sequence? Martin-Gay Interactive Videos See Video 11.2 16. The thirteenth term of the arithmetic sequence -3, 0, 3, c 17. The twenty-fifth term of the arithmetic sequence 20, 18, 16, c 18. The ninth term of the geometric sequence 5, 10, 20, c 19. The fifth term of the geometric sequence 2, -10, 50, c 20. The sixth term of the geometric sequence 1 2 , 3 2 , 9 2 , c Find the indicated term of each sequence. See Examples 4 and 9. 21. The eighth term of the arithmetic sequence whose fourth term is 19 and whose fifteenth term is 52 22. If the second term of an arithmetic sequence is 6 and the tenth term is 30, find the twenty-fifth term. 23. If the second term of an arithmetic progression is -1 and the fourth term is 5, find the ninth term. 24. If the second term of a geometric progression is 15 and the third term is 3, find a1 and r. 25. If the second term of a geometric progression is - 4 3 and the third term is 8 3 , find a1 and r. 26. If the third term of a geometric sequence is 4 and the fourth term is -12, find a1 and r. 27. Explain why 14, 10, and 6 may be the first three terms of an arithmetic sequence when it appears we are subtracting instead of adding to get the next term. 28. Explain why 80, 20, and 5 may be the first three terms of a geometric sequence when it appears we are dividing instead of multiplying to get the next term. MIXED PRACTICE Given are the first three terms of a sequence that is either arithmetic or geometric. If the sequence is arithmetic, find a1 and d. If a se- quence is geometric, find a1 and r. 29. 2, 4, 6 30. 8, 16, 24 31. 5, 10, 20 32. 2, 6, 18 33. 1 2 , 1 10 , 1 50 34. 2 3 , 4 3 , 2 35. x, 5x, 25x 36. y, -3y, 9y 37. p, p + 4, p + 8 38. t, t - 1, t - 2 Find the indicated term of each sequence. 39. The twenty-first term of the arithmetic sequence whose first term is 14 and whose common difference is 1 4 40. The fifth term of the geometric sequence whose first term is 8 and whose common ratio is -3 41. The fourth term of the geometric sequence whose first term is 3 and whose common ratio is - 2 3