466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6-3 Arithmetic and Geometric Sequences
Arithmetic and Geometric Sequences nth-Term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite Geometric Series Sum Formula for Infinite Geometric Series
For most sequences it is difficult to sum an arbitrary number of terms of the sequence without adding term by term. But particular types of sequences, arith- metic sequences and geometric sequences, have certain properties that lead to con- venient and useful formulas for the sums of the corresponding arithmetic series and geometric series. Arithmetic and Geometric Sequences The sequence 5, 7, 9, 11, 13,..., 5 ϩ 2(n Ϫ 1),..., where each term after the first is obtained by adding 2 to the preceding term, is an example of an arithmetic sequence. The sequence 5, 10, 20, 40, 80,..., 5 (2)nϪ1,..., where each term after the first is obtained by multiplying the preceding term by 2, is an example of a geometric sequence.
ARITHMETIC SEQUENCE DEFINITION A sequence
a , a , a ,..., a ,... 1 1 2 3 n is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that
Ϫ ϭ an anϪ1 d
That is,
ϭ ϩ Ͼ an anϪ1 d for every n 1
GEOMETRIC SEQUENCE DEFINITION A sequence
a , a , a ,..., a ,... 2 1 2 3 n is called a geometric sequence, or geometric progression, if there exists a nonzero constant r, called the common ratio, such that 6-3 Arithmetic and Geometric Sequences 467
a n ϭ r DEFINITION anϪ1 2 That is, continued ϭ Ͼ an ranϪ1 for every n 1
Explore/Discuss (A) Graph the arithmetic sequence 5, 7, 9, .... Describe the graphs of all arithmetic sequences with common difference 2. (B) Graph the geometric sequence 5, 10, 20, .... 1 Describe the graphs of all geometric sequences with common ratio 2.
EXAMPLE Recognizing Arithmetic and Geometric Sequences 1 Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence? (A) 1, 2, 3, 5,... (B) Ϫ1, 3, Ϫ9, 27,... (C) 3, 3, 3, 3,... (D) 10, 8.5, 7, 5.5,...
Solutions (A) Since 2 Ϫ 1 Þ 5 Ϫ 3, there is no common difference, so the sequence is 2 Þ 3 not an arithmetic sequence. Since 1 2 , there is no common ratio, so the sequence is not geometric either. (B) The sequence is geometric with common ratio Ϫ3, but it is not arithmetic. (C) The sequence is arithmetic with common difference 0 and it is also geometric with common ratio 1. (D) The sequence is arithmetic with common difference Ϫ1.5, but it is not geometric.
MATCHED PROBLEM Which of the following can be the first four terms of an arithmetic sequence? Of 1 a geometric sequence? (A) 8, 2, 0.5, 0.125,... (B) Ϫ7, Ϫ2, 3, 8,... (C) 1, 5, 25, 100,...
nth-Term Formulas
If {an} is an arithmetic sequence with common difference d, then ϭ ϩ a2 a1 d ϭ ϩ ϭ ϩ a3 a2 d a1 2d ϭ ϩ ϭ ϩ a4 a3 d a1 3d 468 6 SEQUENCES, SERIES, AND PROBABILITY
This suggests Theorem 1, which can be proved by mathematical induction (see Problem 63 in Exercise 6-3).
THE nTH TERM OF AN ARITHMETIC SEQUENCE THEOREM a ϭ a ϩ (n Ϫ 1)d for every n Ͼ 1 1 n 1
Similarly, if {an} is a geometric sequence with common ratio r, then ϭ a2 a1r ϭ ϭ 2 a3 a2r a1r ϭ ϭ 3 a4 a3r a1r
This suggests Theorem 2, which can also be proved by mathematical induction (see Problem 69 in Exercise 6-3).
THE nTH TERM OF A GEOMETRIC SEQUENCE THEOREM a ϭ a r nϪ1 for every n Ͼ 1 2 n 1
EXAMPLE Finding Terms in Arithmetic and Geometric Sequences 2 (A) If the first and tenth terms of an arithmetic sequence are 3 and 30, respec- tively, find the fiftieth term of the sequence. (B) If the first and tenth terms of a geometric sequence are 1 and 4, find the seventeenth term to three decimal places.
ϭ Solutions (A) First use Theorem 1 with a1 3 and Now find a50: ϭ a10 30 to find d: ϭ ϩ Ϫ ϭ ϩ Ϫ an a1 (n 1)d a50 a1 (50 1)3 ؒ ϭ ϩ Ϫ ϭ ϩ a10 a1 (10 1)d 3 49 3 30 ϭ 3 ϩ 9d ϭ 150 d ϭ 3
ϭ ϭ ϭ (B) First let n 10, a1 1, a10 4 and use Theorem 2 to find r.
ϭ nϪ1 an a1r 4 ϭ 1r10Ϫ1 r ϭ 41/9 6-3 Arithmetic and Geometric Sequences 469
Now use Theorem 2 again, this time with n ϭ 17.
ϭ 17 ϭ 1/9 17 ϭ 17/9 ഠ a17 a1r 1 (4 ) 4 13.716
MATCHED PROBLEM (A) If the first and fifteenth terms of an arithmetic sequence are Ϫ5 and 23, 2 respectively, find the seventy-third term of the sequence. 1 1 1 (B) Find the eighth term of the geometric sequence , Ϫ , , . . . . 64 32 16
Sum Formulas for Finite Arithmetic Series
If a1, a2, a3,..., an is a finite arithmetic sequence, then the corresponding series ϩ ϩ ϩ ...ϩ a1 a2 a3 an is called an arithmetic series. We will derive two sim- ple and very useful formulas for the sum of an arithmetic series. Let d be the
common difference of the arithmetic sequence a1, a2, a3,..., an and let Sn denote ϩ ϩ ϩ ...ϩ the sum of the series a1 a2 a3 an. Then ϭ ϩ ϩ ϩ ...ϩ ϩ Ϫ ϩ ϩ Ϫ Sn a1 (a1 d) [a1 (n 2)d] [a1 (n 1)d] Reversing the order of the sum, we obtain ϭ ϩ Ϫ ϩ ϩ Ϫ ϩ ...ϩ ϩ ϩ Sn [a1 (n 1)d] [a1 (n 2)d] (a1 d) a1 Adding the left sides of these two equations and corresponding elements of the right sides, we see that ϭ ϩ Ϫ ϩ ϩ Ϫ ϩ ...ϩ ϩ Ϫ 2Sn [2a1 (n 1)d] [2a1 (n 1)d] [2a1 (n 1)d] ϭ ϩ Ϫ n[2a1 (n 1)d] This can be restated as in Theorem 3.
SUM OF AN ARITHMETIC SERIES—FIRST FORM THEOREM n S ϭ [2a ϩ (n Ϫ 1)d] 3 n 2 1
ϩ Ϫ By replacing a1 (n 1)d with an, we obtain a second useful formula for the sum.
SUM OF AN ARITHMETIC SERIES—SECOND FORM THEOREM n S ϭ (a ϩ a ) 4 n 2 1 n 470 6 SEQUENCES, SERIES, AND PROBABILITY
The proof of the first sum formula by mathematical induction is left as an exercise (see Problem 64 in Exercise 6-3).
EXAMPLE Finding the Sum of an Arithmetic Series 3 Find the sum of the first 26 terms of an arithmetic series if the first term is Ϫ7 and d ϭ 3.
ϭ ϭϪ ϭ Solution Let n 26, a1 7, d 3, and use Theorem 3.
n S ϭ [2a ϩ (n Ϫ 1)d] n 2 1 ϭ 26 Ϫ ϩ Ϫ S26 2 [2( 7) (26 1)3] ϭ 793
MATCHED PROBLEM Find the sum of the first 52 terms of an arithmetic series if the first term is 23 3 and d ϭϪ2.
EXAMPLE Finding the Sum of an Arithmetic Series 4 Find the sum of all the odd numbers between 51 and 99, inclusive. ϭ ϭ Solution First, use a1 51, an 99, and Now use Theorem 4 to find S25: Theorem 1 to find n:
n a ϭ a ϩ (n Ϫ 1)dSϭ (a ϩ a ) n 1 n 2 1 n ϭ ϩ Ϫ ϭ 25 ϩ 99 51 (n 1)2 S25 2 (51 99) n ϭ 25 ϭ 1,875
MATCHED PROBLEM Find the sum of all the even numbers between Ϫ22 and 52, inclusive. 4
EXAMPLE Prize Money 5 A 16-team bowling league has $8,000 to be awarded as prize money. If the last-place team is awarded $275 in prize money and the award increases by the same amount for each successive finishing place, how much will the first- place team receive? 6-3 Arithmetic and Geometric Sequences 471
Solution If a1 is the award for the first-place team, a2 is the award for the second-place team, and so on, then the prize money awards form an arithmetic sequence with ϭ ϭ ϭ n 16, a16 275, and S16 8,000. Use Theorem 4 to find a1.
n S ϭ (a ϩ a ) n 2 1 n ϭ 16 ϩ 8,000 2 (a1 275) ϭ a1 725
Thus, the first-place team receives $725.
MATCHED PROBLEM Refer to Example 5. How much prize money is awarded to the second-place team? 5
Sum Formulas for Finite Geometric Series
If a1, a2, a3,..., an is a finite geometric sequence, then the corresponding series ϩ ϩ ϩ ...ϩ a1 a2 a3 an is called a geometric series. As with arithmetic series, we can derive two simple and very useful formulas for the sum of a geometric
series. Let r be the common ratio of the geometric sequence a1, a2, a3,..., an ϩ ϩ ϩ ...ϩ and let Sn denote the sum of the series a1 a2 a3 an. Then
ϭ ϩ ϩ 2 ϩ 3 ϩ ...ϩ nϪ2 ϩ nϪ1 Sn a1 a1r a1r a1r a1r a1r
Multiply both sides of this equation by r to obtain
ϭ ϩ 2 ϩ 3 ϩ ...ϩ nϪ1 ϩ n rSn a1r a1r a1r a1r a1r
Now subtract the left side of the second equation from the left side of the first, and the right side of the second equation from the right side of the first to obtain
Ϫ ϭ Ϫ n Sn rSn a1 a1r Ϫ ϭ Ϫ n Sn(1 r) a1 a1r
Thus, solving for Sn, we obtain the following formula for the sum of a geomet- ric series:
SUM OF A GEOMETRIC SERIES—FIRST FORM THEOREM a Ϫ a rn S ϭ 1 1 r Þ 1 5 n 1 Ϫ r 472 6 SEQUENCES, SERIES, AND PROBABILITY
ϭ nϪ1 ϭ n Since an a1r , or ran a1r , the sum formula also can be written in the following form:
SUM OF A GEOMETRIC SERIES—SECOND FORM THEOREM a Ϫ ra S ϭ 1 n r Þ 1 6 n 1 Ϫ r
The proof of the first sum formula (Theorem 5) by mathematical induction is left as an exercise (see Problem 70, Exercise 6-3). If r ϭ 1, then
ϭ ϩ ϩ 2 ϩ ...ϩ nϪ1 ϭ Sn a1 a1(1) a1(1 ) a1(1 ) na1
EXAMPLE Finding the Sum of a Geometric Series 6 Find the sum of the first 20 terms of a geometric series if the first term is 1 and r ϭ 2.
ϭ ϭ ϭ Solution Let n 20, a1 1, r 2, and use Theorem 5.
a Ϫ a rn S ϭ 1 1 n 1 Ϫ r Ϫ 1 ؒ 220 1 ϭ ϭ 1,048,575 Calculation using a calculator 1 Ϫ 2
MATCHED PROBLEM Find the sum, to two decimal places, of the first 14 terms of a geometric series if the first term is 1 and r ϭϪ2. 6 64
Sum Formula for Infinite Geometric Series ϭ ϭ 1 Consider a geometric series with a1 5 and r 2. What happens to the sum Sn as n increases? To answer this question, we first write the sum formula in the more convenient form
a Ϫ a rn a a rn S ϭ 1 1 ϭ 1 Ϫ 1 (1) n 1 Ϫ r 1 Ϫ r 1 Ϫ r
ϭ ϭ 1 For a1 5 and r 2,
n ϭ Ϫ 1 Sn 10 10 2 6-3 Arithmetic and Geometric Sequences 473
Thus,
ϭ Ϫ 1 S2 10 10 4
ϭ Ϫ 1 S4 10 10 16
ϭ Ϫ 1 S10 10 10 1,024
ϭ Ϫ 1 S20 10 10 1,048,576
1 n It appears that (2) becomes smaller and smaller as n increases and that the sum gets closer and closer to 10. In general, it is possible to show that, if ԽrԽ Ͻ 1, then r n will get closer and closer to 0 as n increases. Symbolically, r n → 0 as n → ϱ. Thus, the term
n a1r 1 Ϫ r
in equation (1) will tend to 0 as n increases, and Sn will tend to
a1 1 Ϫ r
Ͻ In other words, if ԽrԽ 1, then Sn can be made as close to
a1 1 Ϫ r
as we wish by taking n sufficiently large. Thus, we define the sum of an infinite geometric series by the following formula:
SUM OF AN INFINITE GEOMETRIC SERIES DEFINITION a1 Sϱ ϭ ԽrԽ Ͻ 1 3 1 Ϫ r
If ԽrԽ Ն 1, an infinite geometric series has no sum.
EXAMPLE Expressing a Repeating Decimal as a Fraction 7 Represent the repeating decimal 0.454 545 ...ϭ 0.45 as the quotient of two integers. Recall that a repeating decimal names a rational number and that any rational number can be represented as the quotient of two integers. 474 6 SEQUENCES, SERIES, AND PROBABILITY
Solution 0.45 ϭ 0.45 ϩ 0.0045 ϩ 0.000 045 ϩ ...
ϭ The right side of the equation is an infinite geometric series with a1 0.45 and r ϭ 0.01. Thus,
a1 0.45 0.45 5 Sϱ ϭ ϭ ϭ ϭ 1 Ϫ r 1 Ϫ 0.01 0.99 11
5 Hence, 0.45 and 11 name the same rational number. Check the result by dividing 5 by 11.
MATCHED PROBLEM Repeat Example 7 for 0.818 181 ...ϭ 0.81. 7
EXAMPLE Economy Stimulation 8 A state government uses proceeds from a lottery to provide a tax rebate for property owners. Suppose an individual receives a $500 rebate and spends 80% of this, and each of the recipients of the money spent by this individual also spends 80% of what he or she receives, and this process continues with- out end. According to the multiplier doctrine in economics, the effect of the original $500 tax rebate on the economy is multiplied many times. What is the total amount spent if the process continues as indicated?
Solution The individual receives $500 and spends 0.8(500) ϭ $400. The recipients of this $400 spend 0.8(400) ϭ $320, the recipients of this $320 spend 0.8(320) ϭ $256, and so on. Thus, the total spending generated by the $500 rebate is
400 ϩ 320 ϩ 256 ϩ ...ϭ 400 ϩ 0.8(400) ϩ (0.8)2(400) ϩ ...
ϭ ϭ which we recognize as an infinite geometric series with a1 400 and r 0.8. Thus, the total amount spent is
a1 400 400 Sϱ ϭ ϭ ϭ ϭ $2,000 1 Ϫ r 1 Ϫ 0.8 0.2
MATCHED PROBLEM Repeat Example 8 if the tax rebate is $1,000 and the percentage spent by all recip- 8 ients is 90%.
ϭ Explore/Discuss (A) Find an infinite geometric series with a1 10 whose sum is 1,000. ϭ (B) Find an infinite geometric series with a1 10 whose sum is 6. ϭ (C) Suppose that an infinite geometric series with a1 10 has a sum. 2 Explain why that sum must be greater than 5. 6-3 Arithmetic and Geometric Sequences 475
Answers to Matched Problems ϭ 1 1. (A) The sequence is geometric with r 4, but not arithmetic. (B) The sequence is arithmetic with d ϭ 5, but not geometric. (C) The sequence is neither arithmetic nor geometric. Ϫ Ϫ Ϫ 9 2. (A) 139 (B) 2 3. 1,456 4. 570 5. $695 6. 85.33 7.11 8. $9,000
ϭ ϭ ϭ ϭ EXERCISE 6-3 17. a1 3, a20 117; d ?, a101 ? ϭ ϭ ϭ ϭ 18. a1 7, a8 28; d ?, a25 ? A ϭϪ ϭ ϭ 19. a1 12, a40 22; S40 ? ϭ ϭϪ ϭ In Problems 1 and 2, determine whether the following can be 20. a1 24, a24 28; S24 ? the first three terms of an arithmetic or geometric sequence, 21. a ϭ 1, a ϭ 1; a ϭ ?, S ϭ ? and, if so, find the common difference or common ratio and the 1 3 2 2 11 11 ϭ 1 ϭ 1 ϭ ϭ next two terms of the sequence. 22. a1 6, a2 4; a19 ?, S19 ? ϭ ϭ ϭ 1. (A) Ϫ11, Ϫ16, Ϫ21, . . . (B) 2, Ϫ4, 8, . . . 23. a3 13, a10 55; a1 ? 1 1 1 (C)1, 4, 9,... (D)2, 6, 18 ,... ϭϪ ϭ ϭ 24. a9 12, a13 3; a1 ? 2. (A) 5, 20, 100, . . . (B) Ϫ5, Ϫ5, Ϫ5,...
(C) 7, 6.5, 6, . . . (D) 512, 256, 128, . . . Let a1, a2, a3,..., an,...be a geometric sequence. Find each of the indicated quantities in Problems 25–30. Let a , a , a ,..., a,...be an arithmetic sequence. In 1 2 3 n 25. a ϭ 100, a ϭ 1; r ϭ ? 26. a ϭ 10, a ϭ 30; r ϭ ? Problems 3–10, find the indicated qualities. 1 6 1 10 ϭ ϭϪ ϭ ϭ ϭ ϭ ϭϪ ϭ ϭ ϭ ϭ 27. a1 5, r 2; S10 ? 28. a1 3, r 2; S10 ? 3. a1 5, d 4; a2 ?, a3 ?, a4 ? 29. a ϭ 9, a ϭ 8; a ϭ ?, a ϭ ? ϭϪ ϭ ϭ ϭ ϭ 1 4 3 2 3 4. a1 18, d 3; a2 ?, a3 ?, a4 ? 30. a ϭ 12, a ϭϪ4; a ϭ ?, a ϭ ? ϭϪ ϭ ϭ ϭ 1 4 9 2 3 5. a1 3, d 5; a15 ?, S11 ? 51 40 ϭ ϭ ϭ ϭ ϭϭϩ ϭϭϪ 6. a1 3, d 4; a22 ?, S21 ? 31. S51 ͚ (3k 3) ? 32. S40 ͚ (2k 3) ? kϭ1 kϭ1 ϭ ϭ ϭ 7. a1 1, a2 5; S21 ? 7 7 ϭϭϪ kϪ1 ϭϭk 8. a ϭ 5, a ϭ 11; S ϭ ? 33. S7 ͚ ( 3) ? 34. S7 ͚ 3 ? 1 2 11 kϭ1 kϭ1 ϭ ϭ ϭ 9. a1 7, a2 5; a15 ? 35. Find g(1) ϩ g(2) ϩ g(3) ϩ ...ϩ g(51) if g(t) ϭ 5 Ϫ t. ϭϪ ϭϪ ϭ 10. a1 3, d 4; a10 ? 36. Find f(1) ϩ f(2) ϩ f(3) ϩ ...ϩ f(20) if f(x) ϭ 2x Ϫ 5. ϩ ϩ ...ϩ ϭ 1 x 37. Find g(1) g(2) g(10) if g(x) (2) . Let a1, a2, a3,..., an,...be a geometric sequence. In Problems 11–16, find each of the indicated quantities. 38. Find f(1) ϩ f(2) ϩ ...ϩ f(10) if f(x) ϭ 2x. ϭϪ ϭϪ1 ϭ ϭ ϭ 11. a1 6, r 2; a2 ?, a3 ?, a4 ? 39. Find the sum of all the even integers between 21 and 135. ϭ ϭ 2 ϭ ϭ ϭ 12. a1 12, r 3; a2 ?, a3 ?, a4 ? 40. Find the sum of all the odd integers between 100 and 500. ϭ ϭ 1 ϭ 2 13. a1 81, r 3; a10 ? 41. Show that the sum of the first n odd natural numbers is n , ϭ ϭ 1 ϭ using approximate formulas from this section. 14. a1 64, r 2; a13 ? 42. Show that the sum of the first n even natural numbers is 15. a ϭ 3, a ϭ 2,187, r ϭ 3; S ϭ ? 1 7 7 n ϩ n2, using appropriate formulas from this section. 16. a ϭ 1, a ϭ 729, r ϭϪ3; S ϭ ? 1 7 7 43. Find a positive number x so that Ϫ2 ϩ x Ϫ 6 is a three- term geometric series. B 44. Find a positive number x so that 6 ϩ x ϩ 8 is a three-term geometric series. ϭϪ ϭ ϩ Let a1, a2, a3,..., an,...be an arithmetic sequence. In 45. For a given sequence in which a1 3 and an anϪ1 3, – Ͼ Problems 17 24, find the indicated quantities. n 1, find an in terms of n. 476 6 SEQUENCES, SERIES, AND PROBABILITY
n ϭ 46. For the sequence in Problem 45, find Sn ͚ ak in terms 69. Prove, using mathematical induction, that if {an} is a geo- of n. kϭ1 metric sequence, then ϭ nϪ1 ʦ an a1r n N In Problems 47–50, find the least positive integer n such that Ͻ 70. Prove, using mathematical induction, that if {an} is a geo- an bn by graphing the sequences {an} and {bn} with a graphing utility. Check your answer by using a graphing utility metric sequence, then to display both sequences in table form. a Ϫ a rn S ϭ 1 1 n ʦ N, r Þ 1 n Ϫ ϭ ϩ ϭ n 1 r 47. an 5 8n, bn 1.1 ϭ ϩ ϭ n 71. Given the system of equations 48. an 96 47n, bn 8(1.5) ax ϩ by ϭ c 49. a ϭ 1,000 (0.99)n, b ϭ 2n ϩ 1 n n dx ϩ ey ϭ f ϭ Ϫ ϭ n 50. an 500 n, bn 1.05 where a, b, c, d, e, and f is any arithmetic progression with a nonzero constant difference, show that the system has a In Problems 51–56, find the sum of each infinite geometric unique solution. series that has a sum. 72. The sum of the first and fourth terms of an arithmetic se- ϩ ϩϩ1 ... ϩ ϩ ϩ ... 51. 3 1 3 52. 16 4 1 quence is 2, and the sum of their squares is 20. Find the sum of the first eight terms of the sequence. 53. 2 ϩ 4 ϩ 8 ϩ ... 54. 4 ϩ 6 ϩ 9 ϩ ... ϪϩϪ1 1 ... Ϫ ϩϪ3 ... 55. 2 2 8 56. 21 3 7 APPLICATIONS
In Problems 57–62, represent each repeating decimal as the 73. Business. In investigating different job opportunities, you quotient of two integers. find that firm A will start you at $25,000 per year and guarantee you a raise of $1,200 each year while firm B ϭ ... ϭ ... 57. 0.7 0.7777 58. 0.5 0.5555 will start you at $28,000 per year but will guarantee you a 59. 0.54 ϭ 0.545 454 ... 60. 0.27 ϭ 0.272 727 ... raise of only $800 each year. Over a period of 15 years, how much would you receive from each firm? 61. 3.216 ϭ 3.216 216 216 ... 74. Business. In Problem 73, what would be your annual ϭ ... 62. 5.63 5.636 363 salary at each firm for the tenth year? C 75. Economics. The government, through a subsidy program, distributes $1,000,000. If we assume that each individual or agency spends 0.8 of what is received, and 0.8 of this is 63. Prove, using mathemtical induction, that if {an} is an spent, and so on, how much total increase in spending re- arithmetic sequence, then sults from this government action? a ϭ a ϩ (n Ϫ 1)d for every n Ͼ 1 n 1 76. Economics. Due to reduced taxes, an individual has an
64. Prove, using mathematical induction, that if {an} is an extra $600 in spendable income. If we assume that the in- arithmetic sequence, then dividual spends 70% of this on consumer goods, that the n producers of these goods in turn spend 70% of what they S ϭ [2a ϩ (n Ϫ 1)d] n 2 1 receive on consumer goods, and that this process contin- ues indefinitely, what is the total amount spent on con- ϭϪ ϭϪ Ͼ 65. If in a given sequence, a1 2 and an 3anϪ1, n 1, sumer goods?
find an in terms of n. ★ 77. Business. If $P is invested at r% compounded annually, n ϭ the amount A present after n years forms a geometric pro- 66. For the sequence in Problem 65, find Sn ͚ ak in terms ϭ gression with a common ratio 1 ϩ r. Write a formula for of n. k 1 the amount present after n years. How long will it take a 67. Show that (x2 ϩ xy ϩ y2), (z2 ϩ xz ϩ x2), and (y2 ϩ yz ϩ sum of money P to double if invested at 6% interest com- z2) are consecutive terms of an arithmetic progression if x, pounded annually? y, and z form an arithmetic progression. (From U.S.S.R. ★ 78. Population Growth. If a population of A people grows at Mathematical Olympiads, 1955–1956, Grade 9.) 0 the constant rate of r% per year, the population after t 68. Take 121 terms of each arithmetic progression 2, 7, years forms a geometric progression with a common ratio 12, . . . and 2, 5, 8, .... How many numbers will there be 1 ϩ r. Write a formula for the total population after t in common? (From U.S.S.R. Mathematical Olympiads, years. If the world’s population is increasing at the rate of 1955–1956, Grade 9.) 2% per year, how long will it take to double? 6-3 Arithmetic and Geometric Sequences 477
79. Finance. Eleven years ago an investment earned $7,000 83. Food Chain. A plant is eaten by an insect, an insect by a for the year. Last year the investment earned $14,000. If trout, a trout by a salmon, a salmon by a bear, and the bear the earnings from the investment have increased the same is eaten by you. If only 20% of the energy is transformed amount each year, what is the yearly increase and how from one stage to the next, how many calories must be much income has accrued from the investment over the supplied by plant food to provide you with 2,000 calories past 11 years? from the bear meat?
80. Air Temperature. As dry air moves upward, it expands. ★ 84. Genealogy. If there are 30 years in a generation, how In so doing, it cools at the rate of about 5°F for each many direct ancestors did each of us have 600 years ago? 1,000-foot rise. This is known as the adiabatic process. By direct ancestors we mean parents, grandparents, great- (A) Temperatures at altitudes that are multiples of 1,000 grandparents, and so on. feet form what kind of a sequence? ★ 85. Physics. An object falling from rest in a vacuum near the (B) If the ground temperature is 80°F, write a formula for surface of the Earth falls 16 feet during the first second, 48 the temperature T in terms of n, if n is in thousands n feet during the second second, 80 feet during the third sec- of feet. ond, and so on. 81. Engineering. A rotating flywheel coming to rest rotates (A) How far will the object fall during the eleventh 300 revolutions the first minute (see the figure). If in each second? subsequent minute it rotates two-thirds as many times as (B) How far will the object fall in 11 seconds? in the preceding minute, how many revolutions will the wheel make before coming to rest? (C) How far will the object fall in t seconds?
★ 86. Physics. In Problem 85, how far will the object fall during: (A) The twentieth second? (B) The tth second?
★ 87. Bacteria Growth. A single cholera bacterium divides 1 every 2 hour to produce two complete cholera bacteria. If we start with a colony of A0 bacteria, how many bacteria will we have in t hours, assuming adequate food supply?
★ 88. Cell Division. One leukemic cell injected into a healthy 1 mouse will divide into two cells in about 2 day. At the end of the day these two cells will divide again, with the dou- 1 bling process continuing each 2 day until there are 1 bil- lion cells, at which time the mouse dies. On which day 82. Physics. The first swing of a bob on a pendulum is 10 after the experiment is started does this happen? inches. If on each subsequent swing it travels 0.9 as far as ★★ 89. Astronomy. Ever since the time of the Greek astronomer on the preceding swing, how far will the bob travel before Hipparchus, second century B.C., the brightness of stars coming to rest? has been measured in terms of magnitude. The brightest stars, excluding the sun, are classed as magnitude 1, and the dimmest visible to the eye are classed as magnitude 6. In 1856, the English astronomer N. R. Pogson showed that first-magnitude stars are 100 times brighter than sixth- magnitude stars. If the ratio of brightness between consec-
utive magnitudes is constant, find this ratio. [Hint: If bn is the brightness of an nth-magnitude star, find r for the geo- ϭ metric progression b1, b2, b3,..., given b1 100b6.]
★ 90. Music. The notes on a piano, as measured in cycles per second, form a geometric progression. (A) If A is 400 cycles per second and AЈ, 12 notes higher, is 800 cycles per second, find the constant ratio r. 478 6 SEQUENCES, SERIES, AND PROBABILITY
(B) Find the cycles per second for C, three notes higher tude up to 60 miles, and if the pressure is 15 pounds per than A. square inch at sea level, what will the pressure be 40 miles up?
94. Zeno’s Paradox. Visualize a hypothetical 440-yard oval racetrack that has tapes stretched across the track at the halfway point and at each point that marks the halfway point of each remaining distance thereafter. A runner run- ning around the track has to break the first tape before the second, the second before the third, and so on. From this point of view it appears that he will never finish the race. This famous paradox is attributed to the Greek philoso- pher Zeno (495–435 B.C.). If we assume the runner runs at 91. Puzzle. If you place 1¢ on the first square of a chessboard, 440 yards per minute, the times between tape breakings 2¢ on the second square, 4¢ on the third, and so on, con- form an infinite geometric progression. What is the sum of tinuing to double the amount until all 64 squares are cov- this progression? ered, how much money will be on the sixty-fourth square? How much money will there be on the whole board? 95. Geometry. If the midpoints of the sides of an equilateral triangle are joined by straight lines, the new figure will be an equilateral triangle with a perimeter equal to half the original. If we start with an equilateral triangle with perimeter 1 and form a sequence of “nested” equilateral triangles proceeding as described, what will be the total perimeter of all the triangles that can be formed in this way?
96. Photography. The shutter speeds and f-stops on a camera are given as follows: 1 1 1 1 1 1 1 1 1 Shutter speeds: 1, 2, 4, 8, 15, 30, 60, 125, 250, 500 f-stops: 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22
★ 92. Puzzle. If a sheet of very thin paper 0.001 inch thick is These are very close to being geometric progressions. Es- torn in half, and each half is again torn in half, and this timate their common ratios. process is repeated for a total of 32 times, how high will ★★ 97. Geometry. We know that the sum of the interior angles of the stack of paper be if the pieces are placed one on top of a triangle is 180°. Show that the sums of the interior an- the other? Give the answer to the nearest mile. gles of polygons with 3, 4, 5, 6, ...sides form an arith- ★ 93. Atmospheric Pressure. If atmospheric pressure decreases metic sequence. Find the sum of the interior angles for a roughly by a factor of 10 for each 10-mile increase in alti- 21-sided polygon.
Section 6-4 Multiplication Principle, Permutations, and Combinations
Multiplication Principle Factorial Permutations Combinations
This section introduces some new mathematical tools that are usually referred to as counting techniques. In general, a counting technique is a mathematical method of determining the number of objects in a set without actually enumerat- ing the objects in the set as 1, 2, 3,.... For example, we can count the number