1040 CHAPTER 11 Further Topics in Algebra 11.1 Sequences and Series ■ Sequences ■ Series and Summation Notation ■ Summation Properties and Rules Sequences A sequence is a function that computes an ordered list. For example, the average person in the United States uses 100 gallons of water each day. The function ƒ1n2 = 100n generates the terms of the sequence 100, 200, 300, 400, 500, 600, 700, c , when n = 1, 2, 3, 4, 5, 6, 7, c . This function represents the number of gallons of water used by the average person after n days. As another example, say $100 is deposited into a savings account paying 3% interest compounded annually. The function g1n2 = 10011.032n calculates the account balance after n years. The terms of the sequence are g112, g122, g132, g142, g152, g162, g172, c and can be approximated as 103, 106.09, 109.27, 112.55, 115.93, 119.41, 122.99, c . Sequence A finite sequence is a function that has a set of natural numbers of the form 51, 2, 3, c , n6 as its domain. An infinite sequence has the set of natural numbers as its domain. Instead of using function notation ƒ1x2 to indicate a sequence, it is customary to use an, where an = ƒ1n2. The letter n is used instead of x as a reminder that n represents a natural number. The elements in the range of a sequence, called the terms of the sequence, are a1, a2, a3, c . The elements of both the domain and the range of a sequence are ordered. The first term is found by letting n = 1, the second term is found by letting n = 2, and so on. The general term, or nth term, of the sequence is an. Figure 1 shows graphs of ƒ1x2 = 2x and an = 2n. Notice that ƒ1x2 is a continuous function, and an consists of discrete points. To graph an, we plot points of the form 1n, 2n2 for n = 1, 2, 3, c . We show only the results for n = 1, 2, 3, 4, and 5. Figure 1 (a) −5 0 12 11 The fifth term is 5.2. (b) Figure 2 A graphing calculator can list the terms of a sequence. Using sequence mode to list the first ten terms of the sequence with general term an = n + 1 n produces the result shown in Figure 2(a). The tenth term can be seen by scrolling to the right. Sequences can also be graphed in sequence mode. Figure 2(b) shows a graph of an = n + 1 n . For n = 5, the term is 5 + 1 5 = 5.2. 7 y 5 –5 10 6 2 x f(x) = 2x 0 10 6 2 an 5 –5 n an = 2n 0
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