Mathematics
Geometric Series
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using a specific formula, which depends on the common ratio and the number of terms in the series.
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10 Key excerpts on "Geometric Series"
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
The more general formula is obtained by enlarging the diagram by a scale factor of a, which corresponds to multiplying both sides of the equation by a. Figure 11.3 An infinite geomeii'*. series Sequences and Series 167 Recurring decimals provide an interesting application of Geometric Series. The formula for the sum to infinity can be used to convert the decimal form to the corresponding rational form. For the example of 0.36 shown below the common ratio, r, is 0.01 with 0.36 as the first term, a: The idea of a sequence approaching a limit was discussed in Chapter 7 in conjunction with iterative procedures for solving equations. Infinite Geometric Series provide students with example of the important idea of the limit of a sequence because the sum to infinity is the limit of a sequence of partial sums. Infinite series of many different kinds are of general importance in mathematics. They provide many interesting insights into the curious and fascinating world of the infinite. SIGMA NOTATION AND STATISTICAL FORMULAE Arithmetic sequences provide a simple context for introducing the ideas of sigma notation with E used as a shorthand for the phrase 'the sum of the first n terms'. Extending the notation to include the limits above and below the E symbol only becomes necessary when there might be ambiguity about the number of terms involved in the sum. The sum of the natural numbers, introduced through the triangle numbers in Chapter 3, can be denoted by E/, where i is a 'formula' for the general term: 1+2 + 3 + 4 + 5 + ... + w = Ei = {n(n + 1) This then gives an alternative way of arriving at the sum of an arithmetic sequence. In the first section of this chapter we considered an arithmetic sequence with 5 as the first term and a common difference of 7.
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
Therefore, the sum of this series is S = a 1 _ 1 − r = 1 _ 5 _ 1 − 1 _ 2 = 1 _ 5 _ 1 _ 2 = 2 __ 5 EXAMPLE 8 Show that 0.999 . . . is equal to 1. Solution We begin by writing 0.999 . . . as an infinite Geometric Series. 0.999 . . . = 0.9 + 0.09 + 0.009 + 0.0009 + ⋅ ⋅ ⋅ = 9 __ 10 + 9 ___ 100 + 9 ____ 1,000 + 9 _____ 10,000 + ⋅ ⋅ ⋅ = 9 __ 10 + 9 __ 10 1 __ 10 + 9 __ 10 1 __ 10 2 + 9 __ 10 1 __ 10 3 + ⋅ ⋅ ⋅ As the last line indicates, we have an infinite Geometric Series with a 1 = 9 __ 10 and r = 1 __ 10 . The sum of this series is given by S = a 1 _ 1 − r = 9 __ 10 _ 1 − 1 __ 10 = 9 __ 10 _ 9 __ 10 = 1 GETTING READY FOR CLASS After reading through the preceding section, respond in your own words and in complete sentences. A. What is a common ratio? B. Explain the formula a n = a 1 r n − 1 in words so that someone who wanted to find the nth term of a geometric sequence could do so from your description. C. When is the sum of an infinite Geometric Series a finite number? D. Explain how a repeating decimal can be represented as an infinite Geometric Series. Answers 7. 12 8. S = 7 __ 10 _ 1 − 1 __ 10 = 7 _ 9 7. Find the sum of 4, 8 __ 3 , 16 __ 9 , 32 __ 27 ,… 8. Show that .77777… is equal to 7 __ 9 . 1155 E X E R C I S E S E T 15.3 VOCABULARY REVIEW Choose the correct words to fill in the blanks below. infinite geometric sequence common ratio general term geometric sequence nth partial sum 1. A sequence of numbers in which each term is obtained from the previous term by multiplying by the same amount each time is called a(n) . 2. In a geometric sequence, the amount we multiply each time is called the . 3. The of a geometric sequence written in terms of the first term a 1 and the common ratio r is given by a n = a 1 r n − 1 . 4. The sum of the first n terms of a geometric sequence is called the , denoted by S n and given by S n = a 1 (r n − 1) ______ r − 1 . 5. The sum of a(n) is given by the formula S = a 1 ___ 1 − r .
eBook - PDFCalculus
Single Variable
- Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Adam H. Spiegler(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
An infinite Geometric Series has the form + + 2 + ⋯ + −2 + −1 + + ⋯ . The “⋯” at the end of the second series tells us that the series is going on forever—in other words, that it is infinite. Sum of a Finite Geometric Series The same procedure that enabled us to find the closed form for 10 can be used to find the sum of any finite Geometric Series. Suppose we write for the sum of the first terms, which means up to the term containing −1 : = + + 2 + ⋯ + −2 + −1 . Multiply by : = + 2 + 3 + ⋯ + −1 + . Now subtract from , which cancels out all terms except for two, giving − = − (1 − ) = (1 − ). Provided ≠ 1, we can solve to find a closed form for as follows: The sum of a finite Geometric Series is given by = + + 2 + ⋯ + −1 = (1 − ) 1 − , provided ≠ 1. Note that the value of in the formula for is the number of terms in the sum . Sum of an Infinite Geometric Series In the ampicillin example, we found the sum and then let → ∞. We do the same here. The sum , which shows the effect of the first doses, is an example of a partial sum. The first three partial sums of the series + + 2 + ⋯ + −1 + + ⋯ are 1 = 2 = + 3 = + + 2 . 9.2 Geometric Series 483 To find the sum of this infinite series, we consider the partial sum, , of the first terms. The formula for the sum of a finite Geometric Series gives = + + 2 + ⋯ + −1 = (1 − ) 1 − . What happens to as → ∞? It depends on the value of . If < 1, then → 0 as → ∞, so lim →∞ = lim →∞ (1 − ) 1 − = (1 − 0) 1 − = 1 − . Thus, provided < 1, as → ∞ the partial sums approach a limit of ∕(1 − ). When this happens, we define the sum of the infinite Geometric Series to be that limit and say the series converges to ∕(1 − ).
eBook - PDF- Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
An infinite Geometric Series has the form + + 2 + ⋯ + −2 + −1 + + ⋯ . The “⋯” at the end of the second series tells us that the series is going on forever—in other words, that it is infinite. Sum of a Finite Geometric Series The same procedure that enabled us to find the closed form for 10 can be used to find the sum of any finite Geometric Series. Suppose we write for the sum of the first terms, which means up to the term containing −1 : = + + 2 + ⋯ + −2 + −1 . Multiply by : = + 2 + 3 + ⋯ + −1 + . Now subtract from , which cancels out all terms except for two, giving − = − (1 − ) = (1 − ). Provided ≠ 1, we can solve to find a closed form for as follows: The sum of a finite Geometric Series is given by = + + 2 + ⋯ + −1 = (1 − ) 1 − , provided ≠ 1. Note that the value of in the formula for is the number of terms in the sum . Sum of an Infinite Geometric Series In the ampicillin example, we found the sum and then let → ∞. We do the same here. The sum , which shows the effect of the first doses, is an example of a partial sum. The first three partial sums of the series + + 2 + ⋯ + −1 + + ⋯ are 1 = 2 = + 3 = + + 2 . 9.2 Geometric Series 483 To find the sum of this infinite series, we consider the partial sum, , of the first terms. The formula for the sum of a finite Geometric Series gives = + + 2 + ⋯ + −1 = (1 − ) 1 − . What happens to as → ∞? It depends on the value of . If < 1, then → 0 as → ∞, so lim →∞ = lim →∞ (1 − ) 1 − = (1 − 0) 1 − = 1 − . Thus, provided < 1, as → ∞ the partial sums approach a limit of ∕(1 − ). When this happens, we define the sum of the infinite Geometric Series to be that limit and say the series converges to ∕(1 − ).
eBook - PDFPrecalculus
A Prelude to Calculus
- Sheldon Axler(Author)
- 2016(Publication Date)
- Wiley(Publisher)
In other words, we have the following formula. The case r = 1 is excluded to avoid division by 0. Geometric Series The sum of a geometric sequence with first term b, ratio r 6 = 1 of consecutive terms, and n terms is b · 1 - r n 1 - r . In other words, if r 6 = 1 then b + br + br 2 + · · · + br n-1 = b · 1 - r n 1 - r . Example 6 Suppose tuition during your first year in college is $12,000. You expect tuition to increase 6% per year, and you expect to take five years total to graduate. What is the total amount of tuition you should expect to pay in college? solution Tuition each year is 1.06 times the previous year’s tuition; thus we have a geometric sequence. Using the formula above, the sum of this geometric sequence is Here we have the first term b = 12,000, the ratio r = 1.06 of consecutive terms, and the number of terms n = 5. 12000 · 1 - 1.06 5 1 - 1.06 , which equals 67645.1. Thus you should expect to pay a total of about $67,645 in tuition during five years in college. To express the formula b + br + br 2 + · · · + br n-1 = b · 1 - r n 1 - r in words, first rewrite the right side of the equation as b - br n 1 - r . The expression br n would be the next term if we added one more term to the geometric sequence. Thus we have the following restatement of the formula. This box allows you to think about the formula for a Geometric Series in words instead of symbols. Geometric Series The sum of a finite geometric sequence equals the first term minus what would be the term following the last term, divided by 1 minus the ratio of consecutive terms. Section 6.2 Series 457 Example 7 Evaluate the Geometric Series 5 3 + 5 9 + 5 27 + · · · + 5 3 20 . solution The first term of this Geometric Series is 5 3 . The ratio of consecutive terms is 1 3 . If we added one more term to this Geometric Series, the next term would be 5 3 21 .
eBook - PDF- Deborah Hughes-Hallett, Andrew M. Gleason, Daniel E. Flath, Patti Frazer Lock, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale, Jeff Tecosky-Feldman, Joseph Thrash, Karen R. Rhea, Thomas W. Tucker(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
Chapter 10 Geometric Series CONTENTS 10.1 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Repeated Drug Dosage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Repeated Deposits into a Savings Account . . . . . . . . . . . . . . . . . . . . . . . . . Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sum of a Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sum of an Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Applications to Business and Economics . . . . . . . . . . . . . . . . . . . . Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Present Value of an Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Multiplier Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Applications to the Natural Sciences . . . . . . . . . . . . . . . . . . . . . . . . Steady-State Drug Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulation of Toxins in the Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depletion of Natural Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric Series and Differential Equations . . . . . . . . . . . . . . . . . . . . . . . PROJECTS: Do You Have Any Common Ancestors?, Harrod-Hicks Model of an Expanding National Economy, Probability of Winning in Sports, Medical Case Study: Drug Desensitization Schedule .
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
If this process is continued indefinitely, what is the total area that is colored blue? DISCUSS ■ DISCOVER ■ PROVE ■ WRITE 102. PROVE: Reciprocals of a Geometric Sequence If a 1 , a 2 , a 3 , . . . is a geometric sequence with common ratio r, show that the sequence 1 a 1 , 1 a 2 , 1 a 3 , . . . is also a geometric sequence, and find the common ratio. 103. PROVE: Logarithms of a Geometric Sequence If a 1 , a 2 , a 3 , . . . is a geometric sequence with a common ratio r 0 and a 1 0, show that the sequence log a 1 , log a 2 , log a 3 , . . . is an arithmetic sequence, and find the common difference. 104. PROVE: Exponentials of an Arithmetic Sequence If a 1 , a 2 , a 3 , . . . is an arithmetic sequence with common difference d, show that the sequence 10 a 1 , 10 a 2 , 10 a 3 , . . . is a geometric sequence, and find the common ratio. 12.4 MATHEMATICS OF FINANCE ■ The Amount of an Annuity ■ The Present Value of an Annuity ■ Installment Buying Many financial transactions involve payments that are made at regular intervals. For example, if you deposit $100 each month in an interest-bearing account, what will the value of your account be at the end of 5 years? If you borrow $100,000 to buy a house, how much must your monthly payments be in order to pay off the loan in 30 years? Each of these questions involves the sum of a sequence of numbers; we use the results of the preceding section to answer them here. ■ The Amount of an Annuity An annuity is a sum of money that is paid in regular equal payments. Although the word annuity suggests annual (or yearly) payments, they can be made semiannually, quarterly, monthly, or at some other regular interval. Payments are usually made at the Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
No longer available |Learn more- Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Sequences and Series 235 The examples shown illustrate that geometric progressions grow in magnitude rather quickly for values of r such that | r | . 1. This feature of rapid geometric growth is evident if we consider a famous story from antiquity. EXAMPLE 6.28 Legend has it that a king who loved chess was visited by a sage who challenged the king to a game of chess. The sage requested that, if he were to defeat the king, his reward would be in rice, according to the following plan: on a chess-board, in the first square, a single grain of rice would be placed. In each succes-sive square of the board, the number of rice grains would be doubled until every square of the board was accounted for. The king agreed and (naturally, for what else would you expect in such a legend?) lost the game. When settling accounts was begun, the true calamity of the price of defeat came home to him. What is the quantity of rice committed to by the king as payment? SOLUTION Since the number of grains of rice on each successive square is precisely twice the quantity applied to the preceding square, the number of grains on the suc-cessive squares followed a geometric progression: 1 grain, 2 grains, 4 grains, 8 grains, and so forth. The initial term of the sequence is a 1 5 1, and the common ratio is r 5 2. Since a chessboard has 64 squares, the total number of grains of rice earned by the sage can be calculated using the sum of terms of a geometric sequence: S 64 18 446 744 73 7 9 551 615 H11005 H11005 H11002 H11002 1 1 2 1 2 0 0 64 ( ) , , , , , , grains of rice.
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Find the first three terms. 46. The common ratio is 1 6 and the third term is 18. Find the first and seventh terms. 47. Which Term? The first term of a geometric sequence is 1536 and the common ratio is 1 2 . Which term of the sequence is 6? 48. Which Term? The second and fifth terms of a geometric sequence are 30 and 3750, respectively. Which term of the sequence is 468,750? 49–52 ■ Partial Sums of a Geometric Sequence Find the partial sum S n of the geometric sequence that satisfies the given conditions. 49. a 5, r 2, n 6 50. a 2 3 , r 1 3 , n 4 51. a 3 28, a 6 224, n 6 52. a 2 0.12, a 5 0.00096, n 4 53–58 ■ Partial Sums of a Geometric Sequence Find the sum. 53. 1 3 9 . . . 2187 54. 1 1 2 1 4 1 8 . . . 1 512 55. 15 30 60 . . . 960 56. 5120 2560 1280 . . . 20 57. 1.25 12.5 125 . . . 12,500,000 58. 10800 1080 108 . . . 0.000108 59–64 ■ Partial Sums of a Geometric Sequence Find the sum. 59. a 5 k 1 3 A 1 2 B k 1 60. a 5 k 1 8 A 3 2 B k 1 61. a 6 k 1 5 1 2 2 k 1 62. a 6 k 1 10 1 5 2 k 1 63. a 5 k 1 3 A 2 3 B k 1 64. a 6 k 1 64 A 3 2 B k 1 65–76 ■ Infinite Geometric Series Determine whether the infi-nite Geometric Series is convergent or divergent. If it is conver-gent, find its sum. 65. 1 1 3 1 9 1 27 . . . 66. 1 1 2 1 4 1 8 . . . 67. 1 1 3 1 9 1 27 . . . 68. 2 5 4 25 8 125 . . . 69. 1 3 2 a 3 2 b 2 a 3 2 b 3 . . . 70. 1 3 6 1 3 8 1 3 10 1 3 12 . . . 71. 3 3 2 3 4 3 8 . . . 72. 1 1 1 1 . . . 73. 3 3 1 1.1 2 3 1 1.1 2 2 3 1 1.1 2 3 . . . 74. 100 9 10 3 1 3 10 . . . 75. 1 ! 2 1 2 1 2 ! 2 1 4 . . . 76. 1 2 2 2 2 4 . . . 77–82 ■ Repeated Decimal Express the repeating decimal as a fraction. 77. 0.777 . . . 78. 0.253 79. 0.030303 . . . 80. 2.1125 81. 0.112 82. 0.123123123 . . . Copyright 2016 Cengage Learning.
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
In Exercises 35–38, write the series using summa- tion notation (starting with k = 1). Each series in Exercises 35–38 is either an arithmetic series or a Geometric Series. 35. 2 + 4 + 6 + · · · + 100 solution The k th term of this sequence is 2k. The last term corresponds to k = 50. Thus 2 + 4 + 6 + · · · + 100 = 50 X k=1 2k. 37. 5 9 + 5 27 + 5 81 + · · · + 5 3 40 solution The k th term of this sequence is 5 3 k+1 . The last term corresponds to k = 39 (be- cause when k = 39, the expression 5 3 k+1 equals 5 3 40 ). Thus 5 9 + 5 27 + 5 81 + · · · + 5 3 40 = 39 X k=1 5 3 k+1 . 39. Restate the symbolic version of the formula for evaluating an arithmetic series using summa- tion notation. solution Consider an arithmetic series with n terms, with an initial term b, and with dif- ference d between consecutive terms. The k th term of this series is b + (k - 1)d. Thus the for- mula for evaluating an arithmetic series using summation notation is n X k=1 ( b + (k - 1)d ) = n ( b + (n-1)d 2 ) . This could also be written in the form n-1 X k=0 ( b + kd ) = n ( b + (n-1)d 2 ) . For Exercises 41–44, consider the fable from the beginning of Section 5.4. In this fable, one grain of rice is placed on the first square of a chess- board, then two grains on the second square, then four grains on the third square, and so on, dou- bling the number of grains placed on each square. 41. Find the total number of grains of rice on the first 18 squares of the chessboard. solution The total number of grains of rice on the first 18 squares of the chessboard is 1 + 2 + 4 + 8 + · · · + 2 17 . This is a Geometric Series; the ratio of consecu- tive terms is 2. The term that would follow the last term is 2 18 . Thus the sum of this series is 1 - 2 18 1 - 2 , which equals 2 18 - 1. 43. Find the smallest number n such that the total number of grains of rice on the first n squares of the chessboard is more than 30,000,000.
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