SECTION 12.1 Sequences 859 (a) Algebraic Solution The first term is given as = s 1. 1 To get the second term, use = n 2 in the formula = − s ns n n 1 to get = = ⋅ = s s2 2 1 2. 2 1 To get the third term, use = n 3 in the formula to get = = ⋅ = s s3 3 2 6. 3 2 To get a new term requires knowing the value of the preceding term. The first five terms are = = ⋅ = = ⋅ = = ⋅ = = ⋅ = s s s s s 1 2 1 2 3 2 6 4 6 24 5 24 120 1 2 3 4 5 Do you recognize this sequence? = s n!. n Graphing Solution Using a TI-84 Plus CE First, put the graphing utility into SEQuence mode. Press = Y and enter the recursive formula into the graphing utility to generate the desired sequence. See Figure 8(a). Next, set up the viewing window. Finally, graph the recursive relation and use TRACE to determine the terms in the sequence. See Figure 8(b). For example, the fourth term of the sequence is 24.Table 3 also shows the terms of the sequence. Figure 8 (b) 150 0 6 (b) Table 3 Listing the Terms of a Recursively Defined Sequence List the first five terms of the recursively defined sequence = = = + − − u u u u u 1 1 n n n 1 2 2 1 EXAMPLE 6 Algebraic Solution The first two terms are given. Finding each successive term requires knowing the previous two terms. That is, = = = + = + = = + = + = = + = + = u u u u u u u u u u u 1 1 1 1 2 1 2 3 2 3 5 1 2 3 1 2 4 2 3 5 3 4 Graphing Solution Figure 9 shows the first five terms using Desmos. The sequence given in Example 6 is called the Fibonacci sequence, and the terms of the sequence are called Fibonacci numbers. These numbers appear in a wide variety of applications (see Problems 91–94). Now Work PROBLEMS 37 AND 45 3 Use Summation Notation It is often important to find the sum of the first n terms of a sequence { } a , n namely a a a a n 1 2 3 + + + + Figure 9
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