SECTION 12.1 Sequences 855 OBJECTIVES 1 List the First Several Terms of a Sequence (p. 855) 2 List the Terms of a Sequence Defined by a Recursive Formula (p. 858) 3 Use Summation Notation (p. 859) 4 Find the Sum of a Sequence Algebraically and Using a Graphing Utility (p. 861) 5 Solve Annuity and Amortization Problems Using Recursive Formulas (p. 862) 12.1 Sequences • Functions (Section 2.1, pp. 61–68) • Compound Interest (Section 5.7, pp. 345–351) PREPARING FOR THIS SECTION Before getting started, review the following: DEFINITION Sequence A sequence is a function whose domain is the set of positive integers and whose range is a subset of the real numbers. Figure 1 2, 3, 4, n f(n) 1 2 3 4 (1, 1) 3 2 1 1 – 2 1 – 3 1 – 4 ( ) ) ( ) x y 1 2 3 4 (1, 1) 3 2 1 2, ( ) ( ) ( ) 4, 1 – 2 1 – 3 1 – 4 ( 3, f(x) 5 , x . 0 1 –x f(n) 5 , n a positive integer 1 –n (b) (a) In Words Ellipses are three dots ( . . . ) and means the pattern continues indefinitely. When you hear the word sequence as it is used in the phrase “a sequence of events,” you probably think of a collection of events, one of which happens first, another second, and so on. In mathematics, the word sequence also refers to outcomes that are first, second, and so on. In a sequence, the inputs are … 1, 2, 3, . Because a sequence is a function, it has a graph. Figure 1(a) shows the graph of the function ( ) = > f x x x 1 , 0. If all the points on this graph were removed except those whose x -coordinates are positive integers—that is, if all points were removed except ( ) ( ) ( ) 1, 1 , 2, 1 2 , 3, 1 3 , and so on—the remaining points would be the graph of the sequence ( ) = f n n 1 , as shown in Figure 1(b). Note that n is used to represent the independent variable in a sequence. This serves to remind us that n is a positive integer. 1 List the First Several Terms of a Sequence A sequence is usually represented by listing its values in order. For example, the sequence whose graph is given in Figure 1(b) might be represented as f f f f 1, 2, 3, 4, or 1, 1 2 , 1 3 , 1 4 , … … ( ) ( ) ( ) ( ) The list never ends, as the ellipsis indicates.The numbers in this ordered list are called the terms of the sequence. In dealing with sequences, subscripted letters are used such as a1 to represent the first term, a2 for the second term, a3 for the third term, and so on. JoAnn Trejo JoAnn Trejo studied signal transduction at UC San Diego, which is the process by which extracellular stimuli transmit information to cells so that they respond with the appropriate behavior such as movement, contraction, or cell division. JoAnn credits her success to “coming from a family of very strong-minded women who possess an incredible work ethic.” Now Work the ‘Are You Prepared?’ problems on page 864.
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