Generating Terms of a Sequence

7 Key excerpts on "Generating Terms of a Sequence"

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  10.1.1 Conceptual Recognize patterns in a sequence in order to identify the general term. The word sequence means an order in which one thing follows another in succession. In mathematics, it means the same thing. For example, if we write x, 2x 2 , 3x 3 , 4x 4 , 5x 5 , ?, what would the next term in the sequence be, the one where the question mark now stands? The answer is 6x 6 . 10.1 Sequences and Series SKILLS OBJECTIVES • Find terms of a sequence given the general term. • Apply factorial notation. • Apply recursion formulas. • Use summation (sigma) notation to represent a series and evaluate finite series and infinite series (if possible). CONCEPTUAL OBJECTIVES • Recognize patterns in a sequence in order to identify the general term. • Understand why (mn)! is not equal to m!n!. • Understand why the Fibonacci sequence is a recursion formula. • Understand why all finite series sum and why infinite series may or may not sum. In This Chapter We will first define a sequence and a series and then discuss two particular kinds of sequences and series called arithmetic and geometric. We will then discuss mathematical proof by induction. Finally, we will discuss the Binomial theorem, which offers us an efficient way to perform binomial expansions. • Find the general nth term of a sequence or series. • Evaluate a finite arithmetic series. • Determine if an infinite geometric series converges or diverges. • Prove a mathematical statement using induction. • Use the binomial theorem to expand a binomial raised to a positive integer power. LEARNING OBJECTIVES 10.1 Sequences and Series 899 A Sequence A sequence is a function whose domain is a set of positive integers. The function values, or terms, of the sequence are written as a 1 , a 2 , a 3 , . . . , a n , . . . Rather than using function notation, sequences are usually written with subscript (or index) notation, a subscript . A finite sequence has the domain {1, 2, 3, . . . , n} for some positive integer n.

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  Teaching and Learning Algebra

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  With a sequence it is the number of gaps that is crucial: discussion should reinforce this to explain why n — 1 appears in a general formula. T: Can you give a general rule? A: You do one less than the number of terms times 7 and then add on 5. T: What would that be for the nth term? B: 7(n -1) + 5 T: How can we simplify that? C: 7n - 2 T: What does that tell you about the numbers? D: They are all 2 less than a multiple of 7. This links the 'gaps' method to the more direct ways featured in Chapter 3 by which students will have approached finding an expression like In - 2. However, generating the form l(n -1) + 5 is a useful step in helping students to understand the form of the more general formula a + (n — )d for the n th term of an arithmetic sequence with first term a and common difference d. Such a formula is easy to remember, or indeed to derive when needed, if its structure is understood through relating it readily to a simple numerical example. The two words 'sequence' and 'series' are often confused in everyday language where statements like 'a sequence is a series of terms' are not at all uncommon. The word progres-sion, as in 'arithmetic progression' and 'geometric progression', is another source of confusion because it is often not clear whether it refers to a sequence or a series. It is a word that is best avoided if possible because it is rarely used in conjunction with sequences of other types. The distinction in meaning between the two words sequence and series need to be emphasized so that they are used correctly. It is easier to do this by referring to their meaning in the context of their use rather than by attempting to give formal definitions. Returning to the earlier sequence, the next problem is to find the sum of the corresponding series for a particular number of terms and then to find a general procedure.

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  College Algebra

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter 7 7.1 Arithmetic and Geometric Sequences 7.2 Sums of Terms of Arithmetic and Geometric Sequences 7.3 General Sequences and Series 7.4 Counting Methods 7.5 Probability 7.6 The Binomial Theorem 7.7 Mathematical Induction I n a bicycle race with many participants, you might ask how many different possibilities exist for the first-place and second-place finishes. Determining the possibilities involves a counting technique known as the multiplication principle. See Example 1 in Section 7.4. In this chapter you will study various topics in algebra, including sequences, counting methods, probability, and mathematical induction. These concepts are used in a variety of applications, such as finance, sports, pharmacology, and biology, as well as in advanced courses in mathematics. More Topics in Algebra Outline 7 . 1 542 Chapter 7 More Topics in Algebra Arithmetic and Geometric Sequences In this section, we examine sequences of numbers. They occur in a wide variety of contexts, such as music and biology. In Examples 5 and 6, we investigate some specific applications to music and biology. But first, we introduce notation and definitions for our work with sequences. Definition of a Sequence A sequence is a function f ( n ) whose domain is the set of all nonnegative integers (i.e., n = 0, 1, 2, 3, …) and whose range is a subset of the set of all real numbers. The numbers f (0), f (l), f (2), … are called the terms of the sequence. For a nonnegative number n , it is conventional to denote the term that corresponds to n by a n rather than by f ( n ). We shall use both notations in our discussion. Next we will study some special types of sequences for which a general rule exists for the function f . Arithmetic Sequences A sequence in which we set the starting value a 0 to a certain number and add a fixed number d to any term of the sequence to get the next term is known as an arithmetic sequence .

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  It is also useful in deriving formulas in calculus, statistics, and probability. (3x 2  2y) 5 , Section 1 ◆ Sequences and Series 587 20–1 Sequences and Series Before using sequences and series, we must define some terms. Sequences A sequence is a set of quantities, called terms, which follow each other in a definite order. Each term can be determined either by its position in the sequence or by knowledge of the preceding terms. ◆◆◆ Example 1: Here are some different kinds of sequences. (a) The sequence is called a finite sequence because it has a finite number of terms, n. (b) The sequence is called an infinite sequence. The three dots at the end indicate that the sequence continues indefinitely. (c) The sequence 3, 7, 11, 15, . . . is called an arithmetic sequence, or arithmetic progression (AP), because each term after the first is equal to the sum of the preceding term and a constant. That constant (4, in this example) is called the common difference. (d) The sequence 2, 6, 18, 54, . . . is called a geometric sequence, or geometric progression (GP), because each term after the first is equal to the product of the preceding term and a constant. That constant (3, in this example) is called the common ratio. (e) The sequence 1, 1, 2, 3, 5, 8, . . . is called a Fibonacci sequence. Each term after the first two terms is the sum of the two preceding terms. ◆◆◆ Series A series is the indicated sum of a sequence. 189 ◆◆◆ Example 2: Some different kinds of series are given in this example. (a) The series is called a finite series. It is also called a posi- tive series because all of its terms are positive. (b) The series is an infinite series. It is also called an alternating series because the signs of the terms alternate. (c) The series is an infinite arithmetic series. The terms of this series form an AP. (d) is an infinite, alternating, geometric series. The terms of this series form a GP.

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  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  A sequence can be thought of as a function whose domain is a set of integers. Each term of the sequence is an output value for the function, so   =  (). Example 3 List the first five terms of the sequence   =  (), where  () = 500 − 10. Solution Evaluate  () for  = 1, 2, 3, 4, 5:  1 =  (1) = 500 − 10 ⋅ 1 = 490 and  2 =  (2) = 500 − 10 ⋅ 2 = 480. Similarly,  3 = 470,  4 = 460, and  5 = 450. Arithmetic Sequences You buy a used car that has already been driven 15,000 miles and drive it 8000 miles per year. The odometer registers 23,000 miles 1 year after your purchase, 31,000 miles after 2 years, and so on. The yearly odometer readings form a sequence   whose terms are 15,000, 23,000, 31,000, 39,000, … . 1 www.statista.com/statistics/201182/forecast-of-smartphone-users-in-the-us/, accessed March 15, 2017. 13.1 SEQUENCES 13-3 Each term of the sequence is obtained from the previous term by adding 8000; that is, the difference between successive terms is 8000. A sequence in which the difference between pairs of successive terms is a fixed quantity is called an arithmetic sequence. Example 4 Which of the following sequences are arithmetic? (a) 9, 5, 1, −3, −7 (b) 3, 6, 12, 24, 48 (c) 2, 2 + , 2 + 2, 2 + 3 (d) 10, 5, 0, 5, 10 Solution (a) Each term is obtained from the previous term by subtracting 4. This sequence is arithmetic. (b) This sequence is not arithmetic: each term is twice the previous term. The differences are 3, 6, 12, 24. (c) This sequence is arithmetic:  is added to each term to obtain the next term. (d) This is not arithmetic. The difference between the second and first terms is −5, but the difference between the fifth and fourth terms is 5. We can write a formula for the general term of an arithmetic sequence. Look at the sequence 2, 6, 10, 14, 18, …, in which the terms increase by 4, and observe that  1 = 2  2 = 6 = 2 + 1 ⋅ 4  3 = 10 = 2 + 2 ⋅ 4  4 = 14 = 2 + 3 ⋅ 4.

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  I n a bicycle race with many participants, you might ask how many different possibilities exist for the first-place and second-place finishes. Determining the possibilities involves a counting technique known as the multiplication principle. See Example 1 in Section 10.4. In this chapter you will study various topics in algebra, including sequences, counting methods, probability, and mathematical induction. These concepts are used in a variety of applications, such as finance, sports, pharmacology, and biology, as well as in advanced courses in mathematics. Chapter 10 Outline Number of Possibilities for First Place Number of Possibilities for Second Place 10 9 10.1 Sequences 10.2 Sums of Terms of Sequences 10.3 General Sequences and Series 10.4 Counting Methods 10.5 Probability 10.6 The Binomial Theorem 10.7 Mathematical Induction More Topics in Algebra 10.1 10.1 Sequences 809 Sequences In this section, we examine sequences of numbers. They occur in a wide variety of contexts, such as music and biology. In Examples 5 and 6, we investigate some specific applications to music and biology. But first, we introduce notation and definitions for our work with sequences. Definition of a Sequence A sequence is a function f ( n ) whose domain is the set of all nonnegative integers (i.e., n = 0, 1, 2, 3, …) and whose range is a subset of the set of all real numbers. The numbers f (0), f (l), f (2), … are called the terms of the sequence. For a nonnegative number n , it is conventional to denote the term that corresponds to n by a n rather than by f ( n ). We shall use both notations in our discussion. Next we will study some special types of sequences for which a general rule exists for the function f . Arithmetic Sequences A sequence in which we set the starting value a 0 to a certain number and add a fixed number d to any term of the sequence to get the next term is known as an arithmetic sequence .

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  How to Count

  An Introduction to Combinatorics, Second Edition

  • R.B.J.T. Allenby, Alan Slomson(Authors)
  • 2010(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Recursive functions are studied in the branch of mathematical logic known as recursive function theory or computability theory. Recursive definitions are also allowed in several programming languages. Recurrence relations are classified according to the form of the function f that occurs in the relation. In this chapter we discuss recurrence relations that can be solved using the device of generating functions. Exercises 7.3.1A Find a recurrence relation and initial conditions for the sequence { a n }, where a n is the number of strings of n digits (that is, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) that do not contain consecutive even digits (where 0 counts as an even digit). 7.3.1B Find a recurrence relation and initial conditions for the sequence { a n }, where n is the number of strings of n digits that contain no consecutive 0s, no consecutive 1s, and no consecutive 2s. 7.4 FIBONACCI NUMBERS We have already seen in Section 7.2 how to work out the generating function of the sequence defined by the recurrence relation given by Equation 7.8. We are now going to take this idea further. If we have an explicit formula for the generating function, we may be able to use this to derive a formula for the coefficients in its power series. These coefficients are, of course, just the terms of the sequence in which we are interested. Before discussing the general method we give another illustration in relation to, perhaps, the best known of all sequences defined by a recurrence relation, namely, the Fibonacci numbers . These numbers are named after the Italian mathematician Leonardo of Pisa,* who introduced them in connection with the following problem. * Leonardo of Pisa lived during the late twelfth and early thirteenth century.

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