466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6- 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, arithmetic sequences and geometric sequences, have certain properties that lead to convenient and useful formulas for the sums of the corresponding arithmetic series and geometric series. Arithmetic and Geometric Sequences The sequence 5, 7, 9,,,..., 5 (n ),..., where each term after the first is obtained by adding to the preceding term, is an example of an arithmetic sequence. The sequence 5, 0, 0, 40, 80,..., 5 () n,..., where each term after the first is obtained by multiplying the preceding term by, is an example of a geometric sequence. DEFINITION ARITHMETIC SEQUENCE A sequence a, a, a,..., a n,... is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that a n a n d That is, a n a n d for every n DEFINITION GEOMETRIC SEQUENCE A sequence a, a, a,..., a n,... is called a geometric sequence, or geometric progression, if there exists a nonzero constant r, called the common ratio, such that
6- Arithmetic and Geometric Sequences 467 DEFINITION continued a n a n r That is, a n ra n for every n Explore/Discuss (A) Graph the arithmetic sequence 5, 7, 9,... Describe the graphs of all arithmetic sequences with common difference. (B) Graph the geometric sequence 5, 0, 0,... Describe the graphs of all geometric sequences with common ratio. Solutions Recognizing Arithmetic and Geometric Sequences Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence? (A),,, 5,... (B),, 9, 7,... (C),,,,... (D) 0, 8.5, 7, 5.5,... (A) Since 5, there is no common difference, so the sequence is not an arithmetic sequence. Since, there is no common ratio, so the sequence is not geometric either. (B) The sequence is geometric with common ratio, but it is not arithmetic. (C) The sequence is arithmetic with common difference 0 and it is also geometric with common ratio. (D) The sequence is arithmetic with common difference.5, but it is not geometric. Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence? (A) 8,, 0.5, 0.5,... (B) 7,,, 8,... (C), 5, 5, 00,... nth-term Formulas If {a n } is an arithmetic sequence with common difference d, then a a d a a d a d a 4 a d a d
468 6 SEQUENCES, SERIES, AND PROBABILITY This suggests Theorem, which can be proved by mathematical induction (see Problem 6 in Exercise 6-). THEOREM THE nth TERM OF AN ARITHMETIC SEQUENCE a n a (n )d for every n Similarly, if {a n } is a geometric sequence with common ratio r, then a a r a a r a r a 4 a r a r This suggests Theorem, which can also be proved by mathematical induction (see Problem 69 in Exercise 6-). THEOREM THE nth TERM OF A GEOMETRIC SEQUENCE a n a r n for every n Finding Terms in Arithmetic and Geometric Sequences (A) If the first and tenth terms of an arithmetic sequence are and 0, respectively, find the fiftieth term of the sequence. (B) If the first and tenth terms of a geometric sequence are and 4, find the seventeenth term to three decimal places. Solutions (A) First use Theorem with a and Now find a 50 : a 0 0 to find d: a n a (n )d a 0 a (0 )d 0 9d d a 50 a (50 ) 49 50 (B) First let n 0, a, a 0 4 and use Theorem to find r. a n a r n 4 r 0 r 4 /9
6- Arithmetic and Geometric Sequences 469 Now use Theorem again, this time with n 7. a 7 a r 7 (4 /9 ) 7 4 7/9.76 (A) If the first and fifteenth terms of an arithmetic sequence are 5 and, respectively, find the seventy-third term of the sequence. (B) Find the eighth term of the geometric sequence 64,, 6,.... Sum Formulas for Finite Arithmetic Series If a, a, a,..., a n is a finite arithmetic sequence, then the corresponding series a a a... a n is called an arithmetic series. We will derive two simple and very useful formulas for the sum of an arithmetic series. Let d be the common difference of the arithmetic sequence a, a, a,..., a n and let S n denote the sum of the series a a a... a n. Then S n a (a d)... [a (n )d] [a (n )d] Reversing the order of the sum, we obtain S n [a (n )d] [a (n )d]... (a d) a Adding the left sides of these two equations and corresponding elements of the right sides, we see that S n [a (n )d] [a (n )d]... [a (n )d] n[a (n )d] This can be restated as in Theorem. THEOREM SUM OF AN ARITHMETIC SERIES FIRST FORM S n n [a (n )d] By replacing a (n )d with a n, we obtain a second useful formula for the sum. THEOREM 4 SUM OF AN ARITHMETIC SERIES SECOND FORM n S n (a a 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-). Finding the Sum of an Arithmetic Series Find the sum of the first 6 terms of an arithmetic series if the first term is 7 and d. Solution Let n 6, a 7, d, and use Theorem. n S n [a (n )d] 6 S 6 [( 7) (6 )] 79 Find the sum of the first 5 terms of an arithmetic series if the first term is and d. 4 Finding the Sum of an Arithmetic Series Find the sum of all the odd numbers between 5 and 99, inclusive. Solution First, use a 5, a n 99, and Now use Theorem 4 to find S 5 : Theorem to find n: a n a (n )d n S n (a a n ) 99 5 (n ) 5 S 5 (5 99) n 5,875 4 Find the sum of all the even numbers between and 5, inclusive. 5 Prize Money A 6-team bowling league has $8,000 to be awarded as prize money. If the last-place team is awarded $75 in prize money and the award increases by the same amount for each successive finishing place, how much will the firstplace team receive?
6- Arithmetic and Geometric Sequences 47 Solution If a is the award for the first-place team, a is the award for the second-place team, and so on, then the prize money awards form an arithmetic sequence with n 6, a 6 75, and S 6 8,000. Use Theorem 4 to find a. n S n (a a n ) 6 8,000 (a 75) a 75 Thus, the first-place team receives $75. 5 Refer to Example 5. How much prize money is awarded to the second-place team? Sum Formulas for Finite Geometric Series If a, a, a,..., a n is a finite geometric sequence, then the corresponding series a a a... a n 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 a, a, a,..., a n and let S n denote the sum of the series a a a... a n. Then S n a a r a r a r... a r n a r n Multiply both sides of this equation by r to obtain rs n a r a r a r... a r n a r n 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 S n rs n a a r n S n ( r) a a r n Thus, solving for S n, we obtain the following formula for the sum of a geometric series: THEOREM 5 SUM OF A GEOMETRIC SERIES FIRST FORM a a r n S n r r
47 6 SEQUENCES, SERIES, AND PROBABILITY Since a n a r n, or ra n a r n, the sum formula also can be written in the following form: THEOREM 6 SUM OF A GEOMETRIC SERIES SECOND FORM a ra n S n r r The proof of the first sum formula (Theorem 5) by mathematical induction is left as an exercise (see Problem 70, Exercise 6-). If r, then S n a a () a ( )... a ( n ) na 6 Finding the Sum of a Geometric Series Find the sum of the first 0 terms of a geometric series if the first term is and r. Solution Let n 0, a, r, and use Theorem 5. S n a a r n r 0,048,575 Calculation using a calculator 6 Find the sum, to two decimal places, of the first 4 terms of a geometric series if the first term is and r. 64 Sum Formula for Infinite Geometric Series Consider a geometric series with a 5 and r. What happens to the sum S n as n increases? To answer this question, we first write the sum formula in the more convenient form S n a a r n r a r a r n r () For a 5 and r, S n 0 0 n
6- Arithmetic and Geometric Sequences 47 Thus, S 0 0 4 S 4 0 0 6 S 0 0 0,04 S 0 0 0,048,576 It appears that becomes smaller and smaller as n increases and that the sum gets closer and closer to 0. In general, it is possible to show that, if r, then r n will get closer and closer to 0 as n increases. Symbolically, r n 0 as n. Thus, the term a r n r ( )n in equation () will tend to 0 as n increases, and S n will tend to a r In other words, if a r r, then S n can be made as close to as we wish by taking n sufficiently large. Thus, we define the sum of an infinite geometric series by the following formula: DEFINITION SUM OF AN INFINITE GEOMETRIC SERIES S a r r If r, an infinite geometric series has no sum. 7 Expressing a Repeating Decimal as a Fraction Represent the repeating decimal 0.454 545... 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 0.45 0.0045 0.000 045... The right side of the equation is an infinite geometric series with a 0.45 and r 0.0. Thus, S a r 0.45 0.0 0.45 0.99 5 5 Hence, 0.45 and name the same rational number. Check the result by dividing 5 by. 7 Repeat Example 7 for 0.88 8... 0.8. 8 Solution Economy Stimulation 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 without 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? The individual receives $500 and spends 0.8(500) $400. The recipients of this $400 spend 0.8(400) $0, the recipients of this $0 spend 0.8(0) $56, and so on. Thus, the total spending generated by the $500 rebate is 400 0 56... 400 0.8(400) (0.8) (400)... which we recognize as an infinite geometric series with a 400 and r 0.8. Thus, the total amount spent is S a r 400 0.8 400 0. $,000 8 Repeat Example 8 if the tax rebate is $,000 and the percentage spent by all recipients is 90%. Explore/Discuss (A) Find an infinite geometric series with a 0 whose sum is,000. (B) Find an infinite geometric series with a 0 whose sum is 6. (C) Suppose that an infinite geometric series with a 0 has a sum. Explain why that sum must be greater than 5.
6- Arithmetic and Geometric Sequences 475 Answers to Matched Problems. (A) The sequence is geometric with r 4, but not arithmetic. (B) The sequence is arithmetic with d 5, but not geometric. (C) The sequence is neither arithmetic nor geometric. 9. (A) 9 (B).,456 4. 570 5. $695 6. 85. 7. 8. $9,000 EXERCISE 6- A In Problems and, determine whether the following can be the first three terms of an arithmetic or geometric sequence, and, if so, find the common difference or common ratio and the next two terms of the sequence.. (A), 6,,... (B), 4, 8,... (C), 4, 9,... (D), 6, 8,.... (A) 5, 0, 00,... (B) 5, 5, 5,... (C) 7, 6.5, 6,... (D) 5, 56, 8,... Let a, a, a,..., a n,...be an arithmetic sequence. In Problems 0, find the indicated qualities.. a 5, d 4; a?, a?, a 4? 4. a 8, d ; a?, a?, a 4? 5. a, d 5; a 5?, S? 6. a, d 4; a?, S? 7. a, a 5; S? 8. a 5, a ; S? 9. a 7, a 5; a 5? 0. a, d 4; a 0? Let a, a, a,..., a n,...be a geometric sequence. In Problems 6, find each of the indicated quantities.. a 6, r ; a?, a?, a 4?. a, r ; a?, a?, a 4?. a 8, r ; a 0? 4. a 64, r ; a? 5. a, a 7,87, r ; S 7? 6. a, a 7 79, r ; S 7? B Let a, a, a,..., a n,...be an arithmetic sequence. In Problems 7 4, find the indicated quantities. 7. a, a 0 7; d?, a 0? 8. a 7, a 8 8; d?, a 5? 9. a, a 40 ; S 40? 0. a 4, a 4 8; S 4?. a, a ; a?, S? 6. a, a ; a 9?, S 9?. a, a 0 55; a? 4. a 9, a ; a? Let a, a, a,..., a n,...be a geometric sequence. Find each of the indicated quantities in Problems 5 0. 5. a 00, a 6 ; r? 6. a 0, a 0 0; r? 7. a 5, r ; S 0? 8. a, r ; S 0? 9. a 9, a 4 ; a?, a? 0. a, a 4 ; a?, a? 5 k. S 5 (k )?. S 40 (k )? 7 k 4 8 4 9 40 k 7 k. S 7 ( ) k? 4. S 7 k? 5. Find g() g() g()... g(5) if g(t) 5 t. 6. Find f() f() f()... f(0) if f(x) x 5. 7. Find g() g()... g(0) if g(x). 8. Find f() f()... f(0) if f(x) x. ( )x 9. Find the sum of all the even integers between and 5. 40. Find the sum of all the odd integers between 00 and 500. 4. Show that the sum of the first n odd natural numbers is n, using approximate formulas from this section. 4. Show that the sum of the first n even natural numbers is n n, using appropriate formulas from this section. 4. Find a positive number x so that x 6 is a threeterm geometric series. 44. Find a positive number x so that 6 x 8 is a three-term geometric series. 45. For a given sequence in which a and a n a n, n, find a n in terms of n.
476 6 SEQUENCES, SERIES, AND PROBABILITY 46. For the sequence in Problem 45, find S n a k in terms of n. In Problems 47 50, find the least positive integer n such that a n b n by graphing the sequences {a n } and {b n } with a graphing utility. Check your answer by using a graphing utility to display both sequences in table form. 47. a n 5 8n, b n. n 48. a n 96 47n, b n 8(.5) n 49. a n,000 (0.99) n, b n n 50. a n 500 n, b n.05 n In Problems 5 56, find the sum of each infinite geometric series that has a sum. 5.... 5. 6 4... 5. 4 8... 54. 4 6 9... 55.... 56.... In Problems 57 6, represent each repeating decimal as the quotient of two integers. 57. 0.7 0.7777... 58. 0.5 0.5555... 59. 0.54 0.545 454... 60. 0.7 0.7 77... 6..6.6 6 6... 6. 5.6 5.66 6... C 8 n k 6. Prove, using mathemtical induction, that if {a n } is an arithmetic sequence, then a n a (n )d for every n 64. Prove, using mathematical induction, that if {a n } is an arithmetic sequence, then n S n [a (n )d] 65. If in a given sequence, a and a n a n, n, find a n in terms of n. n k 66. For the sequence in Problem 65, find S n a k in terms of n. 67. Show that (x xy y ), (z xz x ), and (y yz z ) are consecutive terms of an arithmetic progression if x, y, and z form an arithmetic progression. (From U.S.S.R. Mathematical Olympiads, 955 956, Grade 9.) 68. Take terms of each arithmetic progression, 7,,... and, 5, 8,... How many numbers will there be in common? (From U.S.S.R. Mathematical Olympiads, 955 956, Grade 9.) 7 69. Prove, using mathematical induction, that if {a n } is a geometric sequence, then a n a r n n N 70. Prove, using mathematical induction, that if {a n } is a geometric sequence, then a a r n S n n N, r r 7. Given the system of equations ax by c dx ey f where a, b, c, d, e, and f is any arithmetic progression with a nonzero constant difference, show that the system has a unique solution. 7. The sum of the first and fourth terms of an arithmetic sequence is, and the sum of their squares is 0. Find the sum of the first eight terms of the sequence. APPLICATIONS 7. Business. In investigating different job opportunities, you find that firm A will start you at $5,000 per year and guarantee you a raise of $,00 each year while firm B will start you at $8,000 per year but will guarantee you a raise of only $800 each year. Over a period of 5 years, how much would you receive from each firm? 74. Business. In Problem 7, what would be your annual salary at each firm for the tenth year? 75. Economics. The government, through a subsidy program, distributes $,000,000. If we assume that each individual or agency spends 0.8 of what is received, and 0.8 of this is spent, and so on, how much total increase in spending results from this government action? 76. Economics. Due to reduced taxes, an individual has an extra $600 in spendable income. If we assume that the individual spends 70% of this on consumer goods, that the producers of these goods in turn spend 70% of what they receive on consumer goods, and that this process continues indefinitely, what is the total amount spent on consumer goods? 77. Business. If $P is invested at r% compounded annually, the amount A present after n years forms a geometric progression with a common ratio r. Write a formula for the amount present after n years. How long will it take a sum of money P to double if invested at 6% interest compounded annually? 78. Population Growth. If a population of A 0 people grows at the constant rate of r% per year, the population after t years forms a geometric progression with a common ratio r. Write a formula for the total population after t years. If the world s population is increasing at the rate of % per year, how long will it take to double?
6- Arithmetic and Geometric Sequences 477 79. Finance. Eleven years ago an investment earned $7,000 for the year. Last year the investment earned $4,000. If the earnings from the investment have increased the same amount each year, what is the yearly increase and how much income has accrued from the investment over the past years? 80. Air Temperature. As dry air moves upward, it expands. In so doing, it cools at the rate of about 5 F for each,000-foot rise. This is known as the adiabatic process. (A) Temperatures at altitudes that are multiples of,000 feet form what kind of a sequence? (B) If the ground temperature is 80 F, write a formula for the temperature T n in terms of n, if n is in thousands of feet. 8. Engineering. A rotating flywheel coming to rest rotates 00 revolutions the first minute (see the figure). If in each subsequent minute it rotates two-thirds as many times as in the preceding minute, how many revolutions will the wheel make before coming to rest? 8. Food Chain. A plant is eaten by an insect, an insect by a trout, a trout by a salmon, a salmon by a bear, and the bear is eaten by you. If only 0% of the energy is transformed from one stage to the next, how many calories must be supplied by plant food to provide you with,000 calories from the bear meat? 84. Genealogy. If there are 0 years in a generation, how many direct ancestors did each of us have 600 years ago? By direct ancestors we mean parents, grandparents, greatgrandparents, and so on. 85. Physics. An object falling from rest in a vacuum near the surface of the Earth falls 6 feet during the first second, 48 feet during the second second, 80 feet during the third second, and so on. (A) How far will the object fall during the eleventh second? (B) How far will the object fall in seconds? (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 every hour to produce two complete cholera bacteria. If we start with a colony of A 0 bacteria, how many bacteria will we have in t hours, assuming adequate food supply? 8. Physics. The first swing of a bob on a pendulum is 0 inches. If on each subsequent swing it travels 0.9 as far as on the preceding swing, how far will the bob travel before coming to rest? 88. Cell Division. One leukemic cell injected into a healthy mouse will divide into two cells in about day. At the end of the day these two cells will divide again, with the doubling process continuing each day until there are billion cells, at which time the mouse dies. On which day after the experiment is started does this happen? 89. Astronomy. Ever since the time of the Greek astronomer Hipparchus, second century B.C., the brightness of stars has been measured in terms of magnitude. The brightest stars, excluding the sun, are classed as magnitude, and the dimmest visible to the eye are classed as magnitude 6. In 856, the English astronomer N. R. Pogson showed that first-magnitude stars are 00 times brighter than sixthmagnitude stars. If the ratio of brightness between consecutive magnitudes is constant, find this ratio. [Hint: If b n is the brightness of an nth-magnitude star, find r for the geometric progression b, b, b,..., given b 00b 6.] 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 A, 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 than A. 9. Puzzle. If you place on the first square of a chessboard, on the second square, 4 on the third, and so on, continuing to double the amount until all 64 squares are covered, how much money will be on the sixty-fourth square? How much money will there be on the whole board? 9. Puzzle. If a sheet of very thin paper 0.00 inch thick is torn in half, and each half is again torn in half, and this process is repeated for a total of times, how high will the stack of paper be if the pieces are placed one on top of the other? Give the answer to the nearest mile. 9. Atmospheric Pressure. If atmospheric pressure decreases roughly by a factor of 0 for each 0-mile increase in altitude up to 60 miles, and if the pressure is 5 pounds per 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 running 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 philosopher Zeno (495 45 B.C.). If we assume the runner runs at 440 yards per minute, the times between tape breakings form an infinite geometric progression. What is the sum of this progression? 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 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: Shutter speeds:,, 4, 8, 5, 0, 60, 5, 50, 500 f-stops:.4,,.8, 4, 5.6, 8,, 6, These are very close to being geometric progressions. Estimate their common ratios. 97. Geometry. We know that the sum of the interior angles of a triangle is 80. Show that the sums of the interior angles of polygons with, 4, 5, 6,...sides form an arithmetic sequence. Find the sum of the interior angles for a -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 enumerating the objects in the set as,,,... For example, we can count the number