Chapter 12 Sequences; Induction; the Binomial Theorem

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Chapter 12: Sequences; Induction; the Binomial Theorem 3 15. { cn} { 2n } = Examine the terms of the sequence: 2, 16, 54, 128, 250, ... There is no common difference; there is no common ratio; neither. 2 16. { dn} { 2n 1} = − Examine the terms of the sequence: 1, 7, 17, 31, 49, ... There is no common difference; there is no common ratio; neither. 3 17. { } { 2 n n } s = Geometric 3( n+ 1) 3n+ 3 2 2 3n+ 3−3n 3 r = = = 2 = 2 = 8 3n 3n 2 2 ⎛ n n 1−8 ⎞ ⎛1−8 ⎞ 8 n S n = 8 ⎜ = 8 = 8 −1 1 8 ⎟ ⎜ 7 ⎟ ⎝ − ⎠ ⎝ − ⎠ 7 2 18. { } { 3 n n } u = Geometric 2( n+ 1) 2n+ 2 ( ) 3 3 2n+ 2−2n 2 r = = = 3 = 3 = 9 2n 2n 3 3 ⎛ n n 1−9 ⎞ ⎛1−9 ⎞ 9 n S n = 9 ⎜ = 9 = 9 −1 1 9 ⎟ ⎜ 8 ⎟ ⎝ − ⎠ ⎝ − ⎠ 8 ( ) 19. 0, 4, 8, 12, ... Arithmetic d = 4− 0= 4 n n Sn = ( 2(0) + ( n− 1)4 ) = ( 4( n− 1) ) = 2 n( n− 1) 2 2 20. 1, –3, –7, –11, ... Arithmetic d =−3− 1=− 4 21. n n Sn = + n− − = 2 2 − n+ n = (6 − 4 n) = n( 3−2n) 2 ( 2(1) ( 1)( 4) ) ( 2 4 4) 3 3 3 3 3, , , , , ... 2 4 8 16 ⎛3 ⎞ ⎜ ⎟ 2 3 1 1 r = ⎝ ⎠ = ⋅ = 3 2 3 2 S n Geometric ⎛ n n ⎛1⎞ ⎞ ⎛ ⎛1⎞ ⎞ ⎜1−⎜ ⎟ ⎟ ⎜1−⎜ ⎟ ⎟ n 2 2 ⎛ ⎛1 ⎞ ⎞ = 3⎜ ⎝ ⎠ ⎟= 3⎜ ⎝ ⎠ ⎟= 6 1− ⎜ 1 ⎟ ⎜ 1 ⎜ ⎟ 1 ⎛ ⎞ ⎟ ⎜ ⎝2 ⎠ ⎟ − ⎝ ⎠ ⎜ 2 ⎟ ⎜ ⎜ ⎟ ⎝ 2 ⎠ ⎟ ⎝ ⎠ ⎝ ⎠ 22. 5 5 5 5 5, − , , − , , ... Geometric 3 9 27 81 ⎛ 5 ⎞ ⎜− ⎟ 3 5 1 1 r = ⎝ ⎠ =− ⋅ =− 5 3 5 3 S n ⎛ n n ⎛ 1⎞ ⎞ ⎛ ⎛ 1⎞ ⎞ ⎜1−⎜− ⎟ ⎟ ⎜1−⎜− ⎟ ⎟ 3 3 = 5⎜ ⎝ ⎠ ⎟= 5⎜ ⎝ ⎠ ⎟ ⎜ ⎛ 1⎞ ⎟ ⎜ ⎛4⎞ ⎟ 1− − ⎜ ⎜ ⎟ 3 ⎟ ⎜ ⎜ ⎟ 3 ⎟ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ n 15 ⎛ ⎛ 1 ⎞ ⎞ = 1− − 4 ⎜ ⎜ ⎟ ⎝ 3⎠ ⎟ ⎝ ⎠ 23. Neither. There is no common difference or common ratio. 24. 25. 26. 27. 28. 3 5 7 9 11 , , , , , Neither. There is no 2 4 6 8 10 common difference or common ratio. 50 50 k= 1 k= 1 ( + ) ⎛50 50 1 ⎞ 3k = 3 k = 3⎜ ⎟= 3825 ⎝ 2 ⎠ ∑ ∑ 30 ∑ k = 1 k 2 ( + )( ⋅ + ) 30 30 1 2 30 1 = = 9455 6 30 30 30 30 30 ∑ ∑ ∑ ∑ ∑ (3k − 9) = 3k− 9 = 3 k− 9 k= 1 k= 1 k= 1 k= 1 k= 1 ⎛30(30 + 1) ⎞ = 3⎜ ⎟−30(9) ⎝ 2 ⎠ = 1395 − 270 = 1125 40 40 40 ∑ ∑ ∑ ( − 2k + 8) = − 2k+ 8 k= 1 k= 1 k= 1 40 40 ∑ ∑ =− 2 k + 8 k= 1 k= 1 ⎛40(1 + 40) ⎞ =− 2⎜ ⎟+ 40(8) ⎝ 2 ⎠ =− 1640 + 320 =−1320 1274 © 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Chapter 12 Review Exercises 29. 7 ∑ k = 1 10 30. ∑ ( ) k = 1 ⎛ 7 ⎞ ⎛ 7 ⎞ ⎛1⎞ ⎛1⎞ k ⎜1− 1 1 1 ⎜ ⎟ ⎟ ⎜ − 3 1 ⎜ ⎟ ⎟ ⎛ ⎞ ⎜ ⎝ ⎠ ⎟ ⎜ ⎝3⎠ ⎟ ⎜ ⎟ = = ⎝3⎠ 3⎜ 1 ⎟ 3⎜ 2 ⎟ 1− ⎛ ⎞ ⎜ 3 ⎟ ⎜ ⎜ ⎟ 3 ⎟ ⎝ ⎠ ⎝ ⎝ ⎠ ⎠ 1⎛ 1 ⎞ = ⎜1− ⎟ 2 ⎝ 2187 ⎠ 1 2186 1093 = ⋅ = ≈0.49977 2 2187 2187 ( ) ⎛ 10 k 1− −2 ⎞ ⎛1−1024 ⎞ − 2 = − 2⎜ ⎟=−2 ⎜ ⎜ ⎟ 1 −( −2) ⎟ 3 ⎝ ⎠ ⎝ ⎠ 2 =− ( − 1023) = 682 3 31. Arithmetic a = 3, d = 4, a = a + ( n− 1) d 1 n 1 a 9 = 3 + (9 − 1)4 = 3+ 8(4) = 3+ 32 = 35 32. Arithmetic a = 1, d = − 2, a = a + ( n−1) d 1 n 1 a 8 = 1 + (8−1)( − 2) = 1+ 7( − 2) = 1− 14 = −13 33. Geometric 1 n 1 a1 = 1, r = , n = 11; an = a1r − 10 a 11 11−1 10 ⎛ 1 ⎞ ⎛ 1 ⎞ = 1⋅ ⎜ ⎟ = ⎜ ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠ 1 = 10,000,000,000 34. Geometric n a = 1, r = 2, n= 11; a = a r − 1 n 1 11−1 10 ( ) ( ) a 11 = 1⋅ 2 = 2 = 1024 35. Arithmetic a = 2, d = 2, n = 9, a = a + ( n− 1) d 1 n 1 a 9 = 2 + (9− 1) 2 = 2+ 8 2 = 9 2 ≈12.7279 36. Geometric n a = 2, r = 2, n= 9, a = a r − 1 n 1 − ( ) ( ) 9 1 8 a9 = 2 2 = 2 2 = 2⋅16 = 16 2 ≈ 22.6274 1 1 37. 7 1 20 1 a = a + 6d = 31 a = a + 19d = 96; Solve the system of equations: a + 6d = 31 1 a1 + 19d = 96 Subtract the second equation from the first equation and solve for d. − 13d =−65 d = 5 a 1 = 31− 6(5) = 31− 30 = 1 a = a + n− d n ( ) ( n )( ) 1 1 = 1+ −1 5 = 1+ 5n −5 = 5n −4 General formula: { a } = { 5n− 4} 38. 8 1 17 1 n a = a + 7d =− 20 a = a + 16d = − 47; Solve the system of equations: a + 7d =−20 1 a1 + 16d =−47 Subtract the second equation from the first equation and solve for d. − 9d = 27 d = −3 a 1 = −20 −7( − 3) = − 20 + 21 = 1 a = a + n− d n ( ) ( n )( ) 1 1 = 1+ −1 −3 = 1− 3n + 3 =− 3n + 4 General formula: { a } = { − 3n+ 4} 39. 10 1 18 1 a = a + 9d = 0 a = a + 17d = 8; Solve the system of equations: a + 9d = 0 1 a1 + 17d = 8 Subtract the second equation from the first equation and solve for d. − 8d =−8 d = 1 a 1 = − 9(1) = − 9 a = a + n− d n ( ) ( n )( ) 1 1 =− 9+ −1 1 =− 9+ n −1 = n −10 General formula: { a } = { n− 10} n n 1275 © 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Page 1 and 2: Chapter 12 Sequences; Induction; th
Page 3 and 4: Section 12.1: Sequences 34. Answers
Page 5 and 6: Section 12.1: Sequences 16 16 16 2
Page 7 and 8: Section 12.1: Sequences c. Find the
Page 9 and 10: Section 12.1: Sequences (b) ⎛ 0.0
Page 11 and 12: Section 12.1: Sequences 97. a. a 1
Page 13 and 14: Section 12.2: Arithmetic Sequences
Page 19 and 20: Section 12.3: Geometric Sequences;
Page 29 and 30: Section 12.4: Mathematical Inductio
Page 35 and 36: Section 12.5: The Binomial Theorem
Page 37 and 38: Section 12.5: The Binomial Theorem
Page 39: Chapter 12 Review Exercises 50. 8 1
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Page 47 and 48: Chapter 12 Test 2. a1 an an − 1 =
Page 49 and 50: Chapter 12 Test ⎛5⎞ ⎛5⎞ ⎛
Page 51 and 52: Chapter 12 Cumulative Review 3x x 2
Page 53 and 54: Chapter 12 Projects Chapter 12 Proj

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