856 CHAPTER 12 Sequences; Induction; the Binomial Theorem For the sequence ( ) = f n n 1 , this means Figure 5 1 0 0 7 a f a f a f a f a f n n 1 1, 2 1 2 , 3 1 3 , 4 1 4 , , 1 , n 1 2 3 4 … … ( ) ( ) ( ) ( ) ( ) = = = = = = = = = = n first term second term third term fourth term th term In other words, the traditional function notation ( ) f n is typically not used for sequences. For the sequence ( ) = f n n 1 , we have a rule for the nth term, which is = a n 1 , n so it is easy to find any term of the sequence. When a formula for the nth term (sometimes called the general term) of a sequence is known, the entire sequence can be represented by placing braces around the formula for the nth term. For example, the sequence whose nth term is ( ) = b 1 2 n n can be represented by { } ( ) { } = b 1 2 n n or by listing the terms b b b b 1 2 , 1 4 , 1 8 , , 1 2 , n n 1 2 3 … … ( ) = = = = Writing the First Several Terms of a Sequence Write down the first six terms of the following sequence and graph it. { } { } = − a n n 1 n EXAMPLE 1 Algebraic Solution The first six terms of the sequence are = − = = = = = = a a a a a a 1 1 1 0, 1 2 , 2 3 , 3 4 , 4 5 , 5 6 1 2 3 4 5 6 See Figure 2 for the graph. Graphing Solution Figure 3 shows the sequence generated on a TI-84 Plus CE graphing calculator. We can see the first few terms of the sequence on the screen. We could also obtain the terms of the sequence using the TABLE feature. First, put the graphing utility in SEQuence mode. Press Y= and enter the formula for the sequence into the graphing utility. See Figure 4. Set up the table with = TblStart 1 and Tbl 1. Δ = See Table 1. Finally, we can graph the sequence. See Figure 5. Notice that the first term of the sequence is hard to see since it lies on the x-axis.TRACEing the graph allows you to determine the terms of the sequence. Now Work PROBLEM 19 Table 1 Figure 3 Figure 4 Figure 2 { } { } = − a n n 1 n 1 3 5 (1, 0) 1.0 0.6 0.2 0.8 0.4 2 4 6 2, 1 – 2 3, 2 – 3 4, 3 – 4 5, 4 – 5 6, 5 – 6 n an ( ) ( ) ( ) ( ) ( )
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