CWA CH 12 FINAL_97_21 - Chapter 12 SEQUENCES AND SERIES...

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Unformatted text preview: Chapter 12 SEQUENCES AND SERIES 12.1 Geometric Sequences 1. a 1 = 2 ; r = 3 ; n = 4 Since n = 4 ; we must …nd a 1 , a 2 , a 3 , and a 4 with a = a 1 = 2 : a 1 = 2 a 2 = 2(3) 2 ¡ 1 = 2(3) 1 = 2(3) = 6 a 3 = 2(3) 3 ¡ 1 = 2(3) 2 = 2(9) = 18 a 4 = 2(3) 4 ¡ 1 = 2(3) 3 = 2(27) = 54 The …rst four terms of this geometric sequence are 2, 6, 18, and 54. 2. a 1 = 4 ; r = 2 ; n = 5 Since n = 5 ; we must …nd a 1 , a 2 , a 3 , a 4 and a 5 with a = a 1 = 4 : a 1 = 4 a 2 = 4(2) 2 ¡ 1 = 4(2) 1 = 4(2) = 8 a 3 = 4(2) 3 ¡ 1 = 4(2) 2 = 4(4) = 16 a 4 = 4(2) 4 ¡ 1 = 4(2) 3 = 4(8) = 32 a 5 = 4(2) 5 ¡ 1 = 4(2) 4 = 4(16) = 64 The …rst …ve terms of this geometric sequence are 4, 8, 16, 32, and 64. 3. a 1 = 1 2 , r = 4 , n = 4 Since n = 4 ; we must …nd a 1 , a 2 , a 3 , and a 4 with a = a 1 = 1 2 : a 1 = 1 2 a 2 = 1 2 (4) 2 ¡ 1 = 1 2 (4) 1 = 1 2 (4) = 2 a 3 = 1 2 (4) 3 ¡ 1 = 1 2 (4) 2 = 1 2 (16) = 8 a 4 = 1 2 (4) 4 ¡ 1 = 1 2 (4) 3 = 1 2 (64) = 32 The …rst four terms of this geometric sequence are 1 2 , 2, 8, and 32. 4. a 1 = 2 3 ; r = 6 ; n = 3 Since n = 3 ; we must …nd a 1 , a 2 , and a 3 with a = a 1 = 2 3 : a 1 = 2 3 a 2 = 2 3 (6) 2 ¡ 1 = 2 3 (6) 1 = 2 3 (6) = 4 a 3 = 2 3 (6) 3 ¡ 1 = 2 3 (6) 2 = 2 3 (36) = 24 The …rst three terms of this geometric sequence are 2 3 , 4, and 24. 5. a 3 = 6 ; a 4 = 12 ; n = 5 Since n = 5 ; we must …nd a 1 , a 2 , a 3 , a 4 , and a 5 with r = a 4 a 3 = 12 6 = 2 : To …nd a; use a 3 = 6 ; r = 2 ; and n = 3 in the formula. 6 = a (2) 3 ¡ 1 6 = a (2) 2 6 = 4 a 3 2 = 3 2 = a a 1 = 3 2 a 2 = 3 2 (2) 2 ¡ 1 = 3 2 (2) 1 = 3 2 (2) = 3 a 3 = 3 2 (2) 3 ¡ 1 = 3 2 (2) 2 = 3 2 (4) = 6 a 4 = 3 2 (2) 4 ¡ 1 = 3 2 (2) 3 = 3 2 (8) = 12 a 5 = 3 2 (2) 5 ¡ 1 = 3 2 (2) 4 = 3 2 (16) = 24 The …rst …ve terms of this geometric sequence are 3 2 ; 3, 6, 12, and 24. 763 764 Chapter 12 SEQUENCES AND SERIES 6. a 2 = 9 ; a 3 = 3 ; n = 4 Since n = 4 ; we must …nd a 1 , a 2 , a 3 , and a 4 with r = a 3 a 2 = 3 9 = 1 3 : To …nd a; use a 2 = 9 ; r = 1 3 ; and n = 2 in the formula. 9 = a μ 1 3 ¶ 2 ¡ 1 9 = a μ 1 3 ¶ 1 9 = a μ 1 3 ¶ 27 = a a 1 = 27 a 2 = 27 μ 1 3 ¶ 2 ¡ 1 = 27 μ 1 3 ¶ 1 = 27 μ 1 3 ¶ = 9 a 3 = 27 μ 1 3 ¶ 3 ¡ 1 = 27 μ 1 3 ¶ 2 = 27 μ 1 9 ¶ = 3 a 4 = 27 μ 1 3 ¶ 4 ¡ 1 = 27 μ 1 3 ¶ 3 = 27 μ 1 27 ¶ = 1 The …rst four terms of this geometric sequence are 27 ; 9, 3, and 1. 7. a 1 = 4 ; r = 3 Since we want a 5 ; use n = 5 in the formula with a = a 1 = 4 and r = 3 : a 5 = 4(3) 5 ¡ 1 = 4(3) 4 = 4(81) = 324 a n = 4(3) n ¡ 1 8. a 1 = 8 ; r = 4 Since we want a 5 ; use n = 5 in the formula with a = a 1 = 8 and r = 4 : a 5 = 8(4) 5 ¡ 1 = 8(4) 4 = 8(256) = 2048 a n = 8(4) n ¡ 1 9. a 1 = ¡ 3 ; r = ¡ 5 Since we want a 5 ; use n = 5 in the formula with a = a 1 = ¡ 3 and r = ¡ 5 : a 5 = ¡ 3( ¡ 5) 5 ¡ 1 = ¡ 3( ¡ 5) 4 = ¡ 3(625) = ¡ 1875 a n = ¡ 3( ¡ 5) n ¡ 1 10. a 1 = ¡ 4 ; r = ¡ 2 Since we want a 5 ; use n = 5 in the formula with a = a 1 = ¡ 4 and r = ¡ 2 : a 5 = ¡ 4( ¡ 2) 5 ¡ 1 = ¡ 4( ¡...
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This note was uploaded on 10/07/2010 for the course CALC 2302 taught by Professor Yuly during the Fall '08 term at University of Texas at Dallas, Richardson.

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CWA CH 12 FINAL_97_21 - Chapter 12 SEQUENCES AND SERIES...

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