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640 CHAPTER 11 Sequences, Series, and the Binomial Theorem
53. 1 2, - 72 and 1 - 3, - 32 54. 1 10, - 142 and 1 5, - 112
CONCEPT EXTENSIONS
Find the first five terms of each sequence. Round each term after the first to four decimal places.
55. a (^) n =
1 2 n
56. 2 n 2 n + 1
57. a (^) n = a 1 + 1 n b
n
58. a (^) n = a 1 +
n b
n
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 n th Fibonacci term is a (^) n = 1 15
c a 1 + 15 2
b
n
b
n 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 1 x 2 = 1 x - 122 + 3 48. f 1 x 2 = 1 x - 222 + 1 49. f 1 x 2 = 21 x + 422 + 2 50. f 1 x 2 = 31 x - 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, 5 2
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 a (^) n = 5 + 1 n - 12 3. a 1 = 5 + 11 - 123 = 5 Replace n with 1. a 2 = 5 + 12 - 123 = 8 Replace n with 2. a 3 = 5 + 13 - 123 = 11 Replace n with 3. a 4 = 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 a (^) n - a (^) n - 1 = 3 c c n th 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 a 1 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
E X A M P L E 1 Write the first five terms of the arithmetic sequence whose first
term is 7 and whose common difference is 2.
Solution
a 1 = 7 a 2 = 7 + 2 = 9 a 3 = 9 + 2 = 11 a 4 = 11 + 2 = 13 a 5 = 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. a 1 = 7 a 2 = 7 + 2 = 9 or a 2 = a 1 + d a 3 = 9 + 2 = 11 or a 3 = a 2 + d = 1 a 1 + d 2 + d = a 1 + 2 d a 4 = 11 + 2 = 13 or a 4 = a 3 + d = 1 a 1 + 2 d 2 + d = a 1 + 3 d a 5 = 13 + 2 = 15 or a 5 = a 4 + d = 1 a 1 + 3 d 2 + d = a 1 + 4 d T 1 subscript - 12 is multiplier c The pattern on the right suggests that the general term a (^) n of an arithmetic se- quence is given by
a (^) n = a 1 + 1 n - 12 d
General Term of an Arithmetic Sequence The general term a (^) n of an arithmetic sequence is given by a (^) n = a 1 + 1 n - 12 d where a 1 is the first term and d is the common difference.
E X A M P L E 2 Consider the arithmetic sequence whose first term is 3 and whose
common difference is - 5.
a. Write an expression for the general term a (^) n.
b. Find the twentieth term of this sequence.
Solution
a. Since this is an arithmetic sequence, the general term a (^) n is given by a (^) n = a 1 + 1 n - 12 d. Here, a 1 = 3 and d = - 5, so
a (^) n = 3 + 1 n - 121 - 52 Let a 1 = 3 and d = - 5. = 3 - 5 n + 5 Multiply. = 8 - 5 n Simplify.
b. a (^) n = 8 - 5 n
a 20 = 8 - 5 #^20 Let n = 20. = 8 - 100 = - 92
Section 11.2 Arithmetic and Geometric Sequences 643
PRACTICE
E X A M P L E 5 Finding Salary
Donna Theime has an offer for a job starting at $40,000 per year and guaranteeing her a raise of $1600 per year for the next 5 years. Write the general term for the arithmetic sequence that models Donna’s potential annual salaries and find her salary for the fourth year.
Solution The first term, a 1 , is 40,000, and d is 1600. So
a (^) n = 40,000 + 1 n - 12116002 = 38,400 + 1600 n a 4 = 38,400 + 1600 #^4 = 44,
Her salary for the fourth year will be $44,800.
5 A starting salary for a consulting company is $57,000 per year with guar- anteed annual increases of $2200 for the next 4 years. Write the general term for the arithmetic sequence that models the potential annual salaries and find the salary for the third year.
OBJECTIVE
2 Identifying Geometric Sequences
We now investigate a geometric sequence, also called a geometric progression. In the sequence 5, 15, 45, 135, c, each term after the first is the product of 3 and the pre- ceding term. This pattern of multiplying by a constant to get the next term defines a geometric sequence. The constant is called the common ratio because it is the ratio of any term (after the first) to its preceding term.
15 5
f n th term h a^ n a (^) n - 1
previous term h
Geometric Sequence and Common Ratio A geometric sequence is a sequence 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 ratio of the sequence.
The sequence 12, 6, 3,
, cis geometric since each term after the first is the
product of the previous term and
E X A M P L E 6 Write the first five terms of a geometric sequence whose first
term is 7 and whose common ratio is 2.
Solution a 1 = 7
a 2 = 7122 = 14 a 3 = 14122 = 28 a 4 = 28122 = 56 a 5 = 56122 = 112
The first five terms are 7, 14, 28, 56, and 112.
644 CHAPTER 11 Sequences, Series, and the Binomial Theorem
PRACTICE
PRACTICE
E X A M P L E 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 a 1 = 2, the fifth term must be a 1 r^5 -^1 ,
or 2 r^4. We know that r is the common ratio of terms, so r must be
, or - 3. Thus,
a 5 = 2 r^4 a 5 = 21 - 324 = 162
8 Find the seventh term of the geometric sequence whose first three terms are
PRACTICE
E X A M P L E 7 Find the eighth term of the geometric sequence whose first term
is 12 and whose common ratio is
Solution Since this is a geometric sequence, the general term a n is given by
a (^) n = a 1 r n^ -^1
Here a 1 = 12 and r =
, so a (^) n = 12 a
b
n - 1
. Evaluate a (^) n for n = 8.
a 8 = 12 a
b
8 - 1 = 12 a
b
7 = 12 a
b =
7 Find the seventh term of the geometric sequence whose first term is 64 and
whose common ratio is
Notice the general pattern of the terms in Example 6. a 1 = 7 a 2 = 7122 = 14 or a 2 = a 11 r 2 a 3 = 14122 = 28 or a 3 = a 21 r 2 = 1 a 1 #^ r 2 #^ r = a 1 r^2 a 4 = 28122 = 56 or a 4 = a 31 r 2 = 1 a 1 #^ r^22 #^ r = a 1 r^3 a 5 = 56122 = 112 or a 5 = a 41 r 2 = 1 a 1 #^ r^32 #^ r = a 1 r^4 d T 1 subscript - 12 is power The pattern on the right above suggests that the general term of a geometric sequence is given by a (^) n = a 1 r n^ -^1.
General Term of a Geometric Sequence The general term a (^) n of a geometric sequence is given by a (^) n = a 1 r n^ -^1 where a 1 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
646 CHAPTER 11 Sequences, Series, and the Binomial Theorem
1. a 1 = 4; d = 2 2. a 1 = 3; d = 10 3. a 1 = 6; d = - 2 4. a 1 = - 20; d = 3 5. a 1 = 1; r = 3 6. a 1 = - 2; r = 2 7. a 1 = 48; r = 1 2 8. a 1 = 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, a 1 , 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.
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 a 1 and r. 25. If the second term of a geometric progression is -
4 3 and the third term is
8 3 , find a 1 and r.
26. If the third term of a geometric sequence is 4 and the fourth term is - 12, find a 1 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 a 1 and d. If a se- quence is geometric, find a 1 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 , 5 x , 25 x 36. y , - 3 y , 9 y 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
Section 11.2 Arithmetic and Geometric Sequences 647
54. On the first swing, the length of the arc through which a pendulum swings is 50 inches. The length of each successive swing is 80% of the preceding swing. Determine whether this sequence is arithmetic or geometric. Find the length of the fourth swing. 55. Jose takes a job that offers a monthly starting salary of $4000 and guarantees him a monthly raise of $125 dur- ing his first year of training. Find the general term of this arithmetic sequence and his monthly salary at the end of his training. 56. At the beginning of Claudia Schaffer’s exercise program, she rides 15 minutes on the Lifecycle. Each week, she increases her riding time by 5 minutes. Write the general term of this arithmetic sequence, and find her riding time after 7 weeks. Find how many weeks it takes her to reach a riding time of 1 hour. 57. If a radioactive element has a half-life of 3 hours, then x grams of the element dwindles to x 2
grams after 3 hours. If a nuclear reactor has 400 grams of that radioactive element, find the amount of radioactive material after 12 hours.
REVIEW AND PREVIEW
Evaluate. See Section 1.3.
58. 5112 + 5122 + 5132 + 5142
59. 1 3112
1 3122
1 3132
60. 212 - 42 + 313 - 42 + 414 - 42 61. 3 0 + 3 1 + 3 2 + 3 3
62.
1 4112
1 4122
1 4132
63. 8 - 1 8 + 1
8 - 2 8 + 2
8 - 3 8 + 3
CONCEPT EXTENSIONS
Write the first four terms of the arithmetic or geometric sequence, whose first term, a 1 , and common difference, d, or common ratio, r, are given.
64. a 1 = +3720, d = - +268. 65. a 1 = +11,782.40, r = 0. 66. a 1 = 26.8, r = 2. 67. a 1 = 19.652; d = - 0. 68. Describe a situation in your life that can be modeled by a geometric sequence. Write an equation for the sequence. 69. Describe a situation in your life that can be modeled by an arithmetic sequence. Write an equation for the sequence. 42. The fourth term of the arithmetic sequence whose first term is 9 and whose common difference is 5 43. The fifteenth term of the arithmetic sequence 3 2
, 2, 5 2
, c
44. The eleventh term of the arithmetic sequence 2, 5 3 , 4 3 , c 45. The sixth term of the geometric sequence 24, 8, 8 3
, c
46. The eighteenth term of the arithmetic sequence 5, 2, - 1, c 47. If the third term of an arithmetic sequence is 2 and the sev- enteenth term is - 40, find the tenth term. 48. If the third term of a geometric sequence is - 28 and the fourth term is - 56, find a 1 and r.
Solve. See Examples 5 and 10.
49. An auditorium has 54 seats in the first row, 58 seats in the second row, 62 seats in the third row, and so on. Find the general term of this arithmetic sequence and the number of seats in the twentieth row. 50. A triangular display of cans in a grocery store has 20 cans in the first row, 17 cans in the next row, and so on, in an arithmetic sequence. Find the general term and the number of cans in the fifth row. Find how many rows there are in the display and how many cans are in the top row. 51. The initial size of a virus culture is 6 units, and it triples its size every day. Find the general term of the geometric sequence that models the culture’s size. 52. A real estate investment broker predicts that a certain prop- erty will increase in value 15% each year. Thus, the yearly property values can be modeled by a geometric sequence whose common ratio r is 1.15. If the initial property value was $500,000, write the first four terms of the sequence and predict the value at the end of the third year. 53. A rubber ball is dropped from a height of 486 feet, and it continues to bounce one-third the height from which it last fell. Write out the first five terms of this geometric sequence and find the general term. Find how many bounces it takes for the ball to rebound less than 1 foot.
