SECTION 12.1 Sequences 857 Note that in the sequence { } bn in Example 2, the signs of the terms alternate. This occurs when we use factors such as ( ) − + 1 , n 1 which equals 1 if n is odd and −1 if n is even, or ( ) −1 , n which equals −1 if n is odd and 1 if n is even. Listing the First Several Terms of a Sequence List the first six terms of the sequence { } bn and graph it. ( ) { } { } ( ) = − + b n 1 2 n n 1 EXAMPLE 2 Figure 6(b) Algebraic Solution The first six terms of the sequence are ( ) ( ) ( ) ( ) ( ) ( ) = − = = − = − = − = = − = = − + + + b b b b b b 1 2 1 2, 1 2 2 1, 1 2 3 2 3 , 1 2 , 2 5 , 1 3 1 1 1 2 2 1 3 3 1 4 5 6 See Figure 6(a) for the graph. Graphing Solution Figure 6(b) shows the sequence generated on Desmos. Figure 6(a) ( ) { } { } ( ) = − + b n 1 2 n n 1 n 1 3 5 (1, 2) 2 1 –1 2 4 (2, –1) 3, 2 – 3 1 – 3 6, – 1 – 2 4, – 5, 2 – 5 bn 6 ( ) ( ) ( ) ( ) Listing the First Several Terms of a Sequence List the first six terms of the sequence { } cn and graph it. { } = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪ ⎪ ⎫ ⎬ ⎪⎪⎪ ⎭ ⎪⎪ ⎪ c n n n n if is even 1 if is odd n EXAMPLE 3 Solution The first six terms of the sequence are = = = = = = = c c c c c c 1 1 1, 2, 1 3 , 4, 1 5 , 6 1 2 3 4 5 6 See Figure 7 for the graph. Now Work PROBLEM 21 The formula that generates the terms of a sequence is not unique. For example, the terms of the sequence in Example 3 could also be found using d n n 1 n { } { } = ( ) − Sometimes a sequence is indicated by an observed pattern in the first few terms that makes it possible to infer the makeup of the nth term. In the examples that follow, enough terms of the sequence are given so that a natural choice for the nth term is suggested. Figure 7 { } = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎫ ⎬ ⎪⎪ ⎪ ⎭ ⎪⎪ ⎪ c n n n n if is even 1 if is odd n 5 3 1 5 3 1 6 4 2 6 4 2 3, (2, 2) (1, 1) (4, 4) (6, 6) 1 – 3 5, 1 – 5 n cn ( ) ( )
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