Mathematics
Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This means that each term is a constant multiple of the preceding term. Geometric sequences are commonly used in mathematics and have applications in various fields such as finance and physics.
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11 Key excerpts on "Geometric Sequence"
eBook - PDFElementary Mathematical Models
An Accessible Development without Calculus, Second Edition
- Dan Kalman, Sacha Forgoston, Albert Goetz(Authors)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
A final section concerns the mathematical constant ?. 4.1 Properties of Geometric Growth Sequences We begin by formalizing the foregoing description of geometric growth as follows. Geometric Growth Sequence Verbal Definition: In a geomet-ric growth sequence each term is found by multiplying the preced-ing term by a constant. 201 202 Chapter 4. Geometric Growth To produce an example of such a sequence, suppose that 𝑎 0 = 10 and that the constant multiplier is 4 . Succeeding terms are then found by repeatedly multiplying by 4. Depicting this in a diagram, we have 10 ×4 −−→ 40 ×4 −−→ 160 ×4 −−→ 640 ×4 −−→ 2560 ⋯ , showing that 10, 40, 160, 640, 2560, ⋯ is a geometric growth sequence. The rapid in-crease in the size of the terms is a reflection of the multiplier 4 . Indeed, the multiplier has fundamental importance in geometric growth se-quences. We will refer to it so frequently that it is convenient to give it a special name. We call it a growth factor . The word factor indicates multiplication, so growth factor denotes the multiplier that produces the observed amount of growth. Using this ter-minology, the terms in the preceding example increase rapidly because of the size of the growth factor 4. In contrast, with a growth factor of 1.1 , we obtain the sequence 10 ×1.1 −−−→ 11 ×1.1 −−−→ 12.1 ×1.1 −−−→ 13.31 ×1.1 −−−→ 14.641 ⋯ . Observe that these terms are still increasing in size, but much more slowly than in the preceding example. With a positive initial term, a Geometric Sequence whose growth factor exceeds 1 always has increasing terms, and the greater the growth factor, the more rapid the increase. If the growth factor is less than 1 (and still positive), the terms decrease in size. For example, if the initial term is again 10 and the growth factor is 0.8 , we have 10 ×0.8 −−−→ 8 ×0.8 −−−→ 6.4 ×0.8 −−−→ 5.12 ×0.8 −−−→ 4.096 ⋯ . A geometric growth sequence whose terms decrease in this way is sometimes referred to as geometric decay.
eBook - PDFIntermediate Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
As with arithmetic sequences, there is a formula to find the finite sum of a Geometric Sequence. To find the formula, we again work with the basic idea of a series being the sum of the terms of a sequence. Because geometric series have a common ratio, we work with the exponents of the common ratio to develop the formula for the finite sum. Start with the definition of a series. S n 5 a 1 1 a 2 1 a 3 1 a 4 1 g 1 a n Substitute the general form of each Geometric Sequence term. S n 5 a 1 1 a 1 r 1 a 1 r 2 1 a 1 r 3 1 g 1 a 1 r 1n232 1 a 1 r 1n222 1 a 1 r 1n212 Multiply both sides by r. rS n 5 a 1 r 1 a 1 r 2 1 a 1 r 3 1 a 1 r 4 1 g 1 a 1 r 1n222 1 a 1 r 1n212 1 a 1 r n Multiply both sides by 21. 2rS n 5 2a 1 r 2 a 1 r 2 2 a 1 r 3 2 a 1 r 4 2 g 2a 1 r 1n222 2 a 1 r 1n212 2a 1 r n Add this equation to the original series. All the middle terms add to 0. S n 5 a 1 1 a 1 r 1 a 1 r 2 1 a 1 r 3 1 g 1 a 1 r 1n232 1 a 1 r 1n222 1 a 1 r 1n212 2rS n 5 2a 1 r 2 a 1 r 2 2 a 1 r 3 2 a 1 r 4 2 g 2a 1 r 1n222 2 a 1 r 1n212 2 a 1 r n S n 2 rS n 5 a 1 1 0 1 0 1 0 1 0 1 c 10 1 0 2a 1 r n Factor out S n from the left side and a 1 from the right side. S n 2 rS n 5 a 1 2 a 1 r n S n 1 1 2 r 2 5 a 1 1 1 2 r n 2 Divide by 1 2 r, to solve for S n . S n 5 a 1 1 1 2 r n 2 1 2 r Sum of the First n Terms of a Geometric Sequence The finite sum, S n , of the first n terms of a Geometric Sequence, a n , with common ratio r ? 1 can be found by using the formula S n 5 a 1 1 a 2 1 g 1 a n 5 a 1 1 1 2 r n 2 1 2 r Copyright 2019 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). 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.
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
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 . A Identify those sequences that are geometric progressions. For those that are geometric, give the common ratio r . 1. 1, 5, 25, 125, . . . 2. 6, 12, 24, 48, . . . 3. 1 __ 2 , 1 __ 6 , 1 __ 18 , 1 __ 54 , . . . 4. 5, 10, 15, 20, . . . 5. 4, 9, 16, 25, . . . 6. −1, 1 __ 3 , − 1 __ 9 , 1 __ 27 , . . . 7. −2, 4, −8, 16, . . . 8. 1, 8, 27, 64, . . . 9. 4, 6, 8, 10, . . . 10. 1, −3, 9, −27, . . . B C Each of the following problems gives some information about a specific geometric progression. 11. If a 1 = 4 and r = 3, find a n . 12. If a 1 = 5 and r = 2, find a n . 13. If a 1 = −2 and r = − 1 __ 2 , find a 6 . 14. If a 1 = 25 and r = − 1 __ 5 , find a 6 . 15. If a 1 = 3 and r = −1, find a 20 . 16. If a 1 = −3 and r = −1, find a 20 . 17. If a 1 = 10 and r = 2, find S 10 . 18. If a 1 = 8 and r = 3, find S 5 . 19. If a 1 = 1 and r = −1, find S 20 . 20. If a 1 = 1 and r = −1, find S 21 . 21. Find a 8 for 1 __ 5 , 1 __ 10 , 1 __ 20 , . . . 22. Find a 8 for 1 __ 2 , 1 __ 10 , 1 __ 50 , . . . 23. Find S 5 for − 1 __ 2 , − 1 __ 4 , − 1 __ 8 , . . . . 24. Find S 6 for − 1 __ 2 , 1, −2, . . . . 25. Find a 10 and S 10 for √ — 2, 2, 2 √ — 2, . . . . 26. Find a 8 and S 8 for √ — 3 , 3, 3 √ — 3 , . . . . 1156 Chapter 15 Sequences and Series 27. Find a 6 and S 6 for 100, 10, 1, . . . . 28. Find a 6 and S 6 for 100, −10, 1, . . . . 29. If a 4 = 40 and a 6 = 160, find r . 30. If a 5 = 1 __ 8 and a 8 = 1 __ 64 , find r . 31. Given the sequence −3, 6,−12, 24, . . . , find a 8 and S 8 . 32. Given the sequence 4, 2, 1, 1/2, . . . , find a 9 and S 9 .
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
If the first term of a Geometric Sequence is a 1 and the common ratio is r , then the formula that gives the general term a n is a n = a 1 r n − 1 The sum of the first n terms of a Geometric Sequence is given by the formula S n = a 1 ( r n − 1) ________ r − 1 EXAMPLES 1. In the sequence a n = 2 n − 1, a 1 = 2(1) − 1 = 1 a 2 = 2(2) − 1 = 3 a 3 = 2(3) − 1 = 5 resulting in 1, 3, 5, … 2. ∑ i = 3 6 ( − 2) i = ( − 2) 3 + ( − 2) 4 + ( − 2) 5 + ( − 2) 6 = − 8 + 16 + ( − 32) + 64 = 40 3. For the sequence 3, 7, 11, 15, … , a 1 = 3 and d = 4. The general term is a n = 3 + ( n − 1)4 = 4 n − 1 Using this formula to find the tenth term, we have a 10 = 4(10) − 1 = 39 The sum of the first 10 terms is S 10 = 10 __ 2 (3 + 39) = 210 4. For the geometric progression 3, 6, 12, 24, … , a 1 = 3 and r = 2. The general term is a n = 3 ⋅ 2 n − 1 The sum of the first 10 terms is S 10 = 3(2 10 − 1) ________ 2 − 1 = 3,069 790 CHAPTER 10 Sequences and Series The Sum of an Infinite Geometric Series [10.4] If a Geometric Sequence has first term a 1 and common ratio r such that r < 1, then the following is called an infinite geometric series : S = ∑ i = 0 ∞ a 1 r i = a 1 + a 1 r + a 1 r 2 + a 1 r 3 + . . . Its sum is given by the formula S = a 1 ____ 1 − r Factorials [10.5] The notation n ! is called n factorial and is defined to be the product of each consecutive integer from n down to 1. That is, 0! = 1 (By definition) 1! = 1 2! = 2 ⋅ 1 3! = 3 ⋅ 2 ⋅ 1 4! = 4 ⋅ 3 ⋅ 2 ⋅ 1 and so on. Binomial Coefficients [10.5] The notation ( n r ) , or n C r , is called a binomial coefficient and is defined by n C r = n r = n ! ________ r !( n − r )! Binomial coefficients can be found by using the formula above or by Pascal’s Triangle , which is 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 and so on. Binomial Theorem [10.5] If n is a positive integer, then the formula for expanding ( x + y ) n is given by ( x + y ) n = ∑ r = 0 n n r x n − r y r = ∑ r = 0 n n C r x n − r y r 5.
eBook - PDF- Ed Wheeler, Jim Brawner(Authors)
- 2013(Publication Date)
- Information Age Publishing(Publisher)
Can a sequence be both arithmetic and geometric? If so, find one. If not, explain why not. 18. Suppose a sequence has first term 2 and third term 18. (a) If the sequence is arithmetic, find the common difference d , and list the first five terms of the sequence. (b) If the sequence is geometric, find two possible values for the common ratio r , and list the first five terms of the Geometric Sequence in each case. 19. There are 1600 students currently enrolled in the Idaville School District. If the enrollment increases by 5% each year, approximately how many students will be enrolled 5 years from now? 20. Alice starts a new job with an annual salary of $40,000 and a 5% raise each year. At the same time, Bob starts a new job with an annual salary of $50,000 and a $2000 raise each year. Will Alice’s annual salary ever be greater than Bob’s? If so, when? 8 4 C H A P T E R 2 : S E T S , F U N C T I O N S , A N D S E Q U E N C E S 21. A chessboard has 64 squares. Suppose a cent is placed on the first square, two cents on the second square, four cents on the third square, eight cents on the fourth square, and twice as many cents on each successive square. (a) Which will be the first square to have more than a million dollars on it? (b) How much money will be on the last square? S E C T I O N 2 . 5 : R E C U R S I V E S E Q U E N C E S 8 5 2.5 Recursive Sequences With many of the sequences we studied in the previous section, we can describe the apparent pattern by saying how to get from one term to the next. For example, we could tell a student how to write down the arithmetic sequence 1, 4, 7, 10, 13, … with only two instructions: 1. Start with 1. 2. Add 3 to any term to get the next term. Notice that both instructions are necessary if we want the student to write down the particular sequence 1, 4, 7, 10, 13, … .
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 - PDFLearning and Teaching Mathematics using Simulations
Plus 2000 Examples from Physics
- Dieter Röss(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
4 Sequences of numbers and series 4.1 Sequences and series By Sequence repeated application of the same arithmetic operations on an initial number A , one creates a logically connected sequence of numbers, which show interesting properties (to guess the formation law of a sequence and thus to continue the initial numbers of a given sequence is a popular type of puzzle). In the following the letters m; n; i; j are used to indicate the position of terms in sequences. They can be 0 or positive integers. If there is no upper limit for the number of terms in a sequence or for the terms in a series ( m ! 1 ), we refer to an infinite sequence or series. 4.1.1 Sequence and series of the natural numbers The particularly simple arithmetic sequence of the natural numbers is created via the repeated addition of the unit 1 ; the individual term is characterized by the lower index ( 1; 2; : : : ), which itself is an increasing natural number. A 1 D 1 I A n C 1 D A n C 1 for n 1 ! A n D 1; 2; 3; 4; 5; 6; : : : : We now define the difference quotient for the terms of an arbitrary sequence with different indices i and j . This number is a measure for the change between two terms with different indices and thus for the growth of the sequence in the interval given by the indices: ĩA i;j D A i A j I ĩ i;j D i j difference quotient: ĩA i;j ĩ i;j D A i A j i j : For consecutive terms, the index interval is 1 and the difference quotient is equal to the difference between the terms: ĩA i;i 1 D A i A i 1 I ĩ i;i 1 D i .i 1/ D 1 difference quotient: ĩA i;i 1 ĩ i;i 1 D A i A i 1 : 4.1 Sequences and series 36 For the sequence of the natural numbers, the difference between consecutive terms is constant and equal to 1 . Therefore their difference quotient is also constant and equal to 1 . ĩA i;i 1 D A i A i 1 D 1 ! difference quotient D 1: The arithmetic sequence has constant growth of consecutive terms.
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).
eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
. . 20. e 2 , e 4 , e 6 , e 8 , . . . 21. 1.0, 1.1, 1.21, 1.331, . . . 22. 1 2 , 1 4 , 1 6 , 1 8 , . . . 23–28 ■ Geometric Sequence? Find the first five terms of the sequence, and determine whether it is geometric. If it is geomet-ric, find the common ratio, and express the n th term of the sequence in the standard form a n ar n 1 . 23. a n 2 1 3 2 n 24. a n 4 3 n 25. a n 1 4 n 26. a n 1 1 2 n 2 n 27. a n ln 1 5 n 1 2 28. a n n n 29–38 ■ Terms of a Geometric Sequence Determine the common ratio, the fifth term, and the n th term of the Geometric Sequence. 29. 2, 6, 18, 54, . . . 30. 7, 14 3 , 28 9 , 56 27 , . . . 31. 0.3, 0.09, 0.027, 0.0081, . . . 32. 1, ! 2 , 2, 2 ! 2 , . . . 33. 144, 12, 1, 1 12 , . . . 34. 8, 2, 1 2 , 1 8 , . . . 35. 3, 3 5 / 3 , 3 7 / 3 , 27, . . . 36. t , t 2 2 , t 3 4 , t 4 8 , . . . 37. 1, s 2 / 7 , s 4 / 7 , s 6 / 7 , . . . 38. 5, 5 c 1 , 5 2 c 1 , 5 3 c 1 , . . . 39–46 ■ Finding Terms of a Geometric Sequence Find the indicated term(s) of the Geometric Sequence with the given description. 39. The first term is 15 and the second term is 6. Find the fourth term. 40. The first term is 1 12 and the second term is 1 2 . Find the sixth term. 41. The third term is 1 3 and the sixth term is 9. Find the first and second terms. 42. The fourth term is 12 and the seventh term is 32 9 . Find the first and n th terms. 43. The third term is 18 and the sixth term is 9216. Find the first and n th terms. 44. The third term is 54 and the sixth term is 729 256 . Find the first and second terms. 45. The common ratio is 0.75 and the fourth term is 729. 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.
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 .
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.
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