Chapter 12 Sequences; Induction; the Binomial Theorem

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Chapter 12: Sequences; Induction; the Binomial Theorem 20. 21. ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ( x+ 3) = ⎜ ⎟x + ⎜ ⎟x (3) + ⎜ ⎟x (3) + ⎜ ⎟x (3) + ⎜ ⎟x (3) + ⎜ ⎟x (3) ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 5 5 4 3 2 2 3 1 4 0 5 5 4 3 2 = x + 5 x (3) + 10x ⋅ 9 + 10 x (27) + 5x⋅ 81+ 243 5 4 3 2 = x + 15x + 90x + 270x + 405x+ 243 4 ⎛4⎞ 4 ⎛4⎞ 3 ⎛4⎞ 2 ⎛4⎞ ⎛4⎞ (3x+ 1) = ⎜ ⎟(3 x) + ⎜ ⎟(3 x) + ⎜ ⎟(3 x) + ⎜ ⎟(3 x) + ⎜ ⎟ ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ 4 3 2 4 3 2 = 81x + 4⋅ 27x + 6⋅ 9x + 4⋅ 3x+ 1 = 81x + 108x + 54x + 12x+ 1 22. ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ (2x+ 3) = ⎜ ⎟(2 x) + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟(2 x) ⋅ 3 + ⎜ ⎟⋅2x⋅ 3 + ⎜ ⎟⋅3 ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 5 5 4 3 2 2 3 4 5 5 4 3 2 = 32x + 5⋅16x ⋅ 3 + 10⋅8x ⋅ 9 + 10⋅4x ⋅ 27 + 5⋅2x⋅ 81+ 243 5 4 3 2 = 32x + 240x + 720x + 1080x + 810x+ 243 2 2 ⎛5⎞ 2 ⎛5⎞ 2 2 ⎛5⎞ 2 2 ⎛5⎞ 2 2 ⎛5⎞ 2 2 ⎛5⎞ 2 23. ( x + y ) = ⎜ ⎟( x ) + ⎜ ⎟( x ) y + ⎜ ⎟( x ) ( y ) + ⎜ ⎟( x ) ( y ) + ⎜ ⎟x ( y ) + ⎜ ⎟( y ) 5 5 4 3 2 2 3 4 5 ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 10 8 2 6 4 4 6 2 8 10 = x + 5x y + 10x y + 10x y + 5x y + y 2 2 ⎛6⎞ 2 ⎛6⎞ 2 2 ⎛6⎞ 2 2 ⎛6⎞ 2 2 ⎛6⎞ 2 2 24. ( x − y ) = ⎜ ⎟( x ) + ⎜ ⎟( x ) ( − y ) + ⎜ ⎟( x ) ( − y ) + ⎜ ⎟( x ) ( − y ) + ⎜ ⎟( x ) ( −y ) 6 6 5 4 2 3 3 2 4 ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎛6⎞ 2 2 5 ⎛6⎞ 2 6 + ⎜ ⎟x ( − y ) + ⎜ ⎟( −y ) ⎝5⎠ ⎝6⎠ 12 10 2 8 4 6 6 4 8 2 10 12 = x − 6x y + 15x y − 20x y + 15x y − 6x y + y ⎛6⎞ ⎛6⎞ ⎛6⎞ ⎛6⎞ 25. ( x + 2) = ⎜ ⎟( x) + ⎜ ⎟( x) ( 2) + ⎜ ⎟( x) ( 2) + ⎜ ⎟( x) ( 2) 6 6 5 1 4 2 3 3 ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎛6⎞ 2 4 ⎛6⎞ 5 ⎛6⎞ 6 + ⎜ ⎟( x) ( 2) + ⎜ ⎟( x)( 2) + ⎜ ⎟( 2) ⎝4⎠ ⎝5⎠ ⎝6⎠ 3 5/2 2 3/2 1/2 = x + 6 2x + 15⋅ 2x + 20⋅ 2 2x + 15⋅ 4x+ 6⋅ 4 2x + 8 3 5/2 2 3/2 1/2 = x + 6 2x + 30x + 40 2x + 60x+ 24 2x + 8 ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ 26. ( x − 3) = ⎜ ⎟( x) + ⎜ ⎟( x) ( − 3) + ⎜ ⎟( x) ( − 3) + ⎜ ⎟( x)( − 3) + ⎜ ⎟( − 3) 4 4 3 1 2 2 3 4 ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ 2 3/2 1/2 = x − 4 3x + 63 ⋅ x−43 ⋅ 3x + 9 2 3/2 1/2 = x − 4 3x + 18x− 12 3x + 9 ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 5 5 4 3 2 2 3 4 5 27. ( ax + by) = ( ax) + ( ax) ⋅ by + ( ax) ( by) + ( ax) ( by) + ax( by) + ( by) 5 5 4 4 3 3 2 2 2 2 3 3 4 4 5 5 = a x + 5a x by+ 10a x b y + 10a x b y + 5axb y + b y ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎛4⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ 4 4 3 2 2 3 4 28. ( ax − by) = ( ax) + ( ax) ( − by) + ( ax) ( − by) + ( ax)( − by) + ( −by) 4 4 3 3 2 2 2 2 3 3 4 4 = a x − 4a x by + 6a x b y − 4axb y + b y 1270 © 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.
Section 12.5: The Binomial Theorem 29. n = 10, j = 4, x = x, a = 3 ⎛10⎞ 10! 10987 ⋅ ⋅ ⋅ ⎜ ⎟x ⋅ 3 = ⋅ 81x = ⋅81x ⎝ 4 ⎠ 4!6! 4⋅3⋅2⋅1 6 = 17,010x 6 The coefficient of x is 17,010. 6 4 6 6 30. n= 10, j = 7, x = x, a = − 3 ⎛10⎞ 3 7 10! 3 ⎜ ⎟x ⋅− ( 3) = ⋅− ( 2187) x ⎝ 7 ⎠ 7!3! 10⋅9⋅8 = ⋅( −2187) x 321 ⋅ ⋅ 3 =−262,440x 3 The coefficient of x is − 262,440. 31. n = 12, j = 5, x = 2 x, a = − 1 ⎛12⎞ 7 5 12! 7 ⎜ ⎟(2 x) ⋅− ( 1) = ⋅128 x ( −1) ⎝ 5 ⎠ 5!7! 12⋅11⋅10⋅9⋅8 = ⋅ ( − 128) x 54321 ⋅ ⋅ ⋅ ⋅ 7 =−101,376x 7 The coefficient of x is − 101,376. 32. n= 12, j = 9, x = 2 x, a = 1 ⎛12⎞ 12! 12⋅11⋅10 ⎜ ⎟(2 x) ⋅ (1) = ⋅ 8 x (1) = ⋅8x ⎝ 9 ⎠ 9!3! 3⋅2⋅1 3 = 1760x 3 The coefficient of x is 1760. 3 9 3 3 33. n= 9, j = 2, x = 2 x, a = 3 ⎛9⎞ 7 2 9! 7 ⎜ ⎟(2 x) ⋅ 3 = ⋅128 x (9) ⎝2⎠ 2!7! 98 ⋅ 128 7 = ⋅ x ⋅ 9 21 ⋅ 7 = 41,472x 7 The coefficient of x is 41,472. 34. n= 9, j = 7, x = 2 x, a = − 3 ⎛9⎞ 2 7 9! 2 ⎜ ⎟(2 x) ⋅− ( 3) = ⋅4 x ( −2187) ⎝7⎠ 7!2! 98 ⋅ 4 2 = ⋅ x ⋅− 2187 21 ⋅ 2 =−314,928x 2 The coefficient of x is − 314,928. 3 7 35. n= 7, j = 4, x = x, a = 3 ⎛7⎞ 7! 7⋅6⋅5 ⎜ ⎟x ⋅ 3 = ⋅ 81x = ⋅ 81x = 2835x ⎝4⎠ 4!3! 3⋅2⋅1 3 4 3 3 3 36. n= 7, j = 2, x = x, a = − 3 ⎛7⎞ 7! 7⋅6 ⎜ ⎟x ⋅− ( 3) = ⋅ 9x = ⋅ 9x = 189x ⎝2⎠ 2!5! 2⋅1 5 2 5 5 5 37. n= 9, j = 2, x = 3 x, a =− 2 ⎛9⎞ 7 2 9! 7 ⎜ ⎟(3 x) ⋅− ( 2) = ⋅2187x ⋅4 ⎝2⎠ 2!7! 98 ⋅ = ⋅ 8748 x = 314,928 x 21 ⋅ 38. n= 8, j = 5, x = 3 x, a = 2 7 7 ⎛8⎞ 3 5 8! 3 ⎜ ⎟(3 x) ⋅ ( 2) = ⋅27x ⋅32 ⎝5⎠ 5!3! 876 ⋅ ⋅ = ⋅ 864 x = 48,384 x 321 ⋅ ⋅ 0 39. The x term in ∑ 12 12 ⎛12⎞ 12− j ⎛1 ⎞ 12 ( ) j ⎛ ⎞ ⎜ ⎟ x x j= 0 j ⎜ ⎟ = ∑ ⎜ ⎟ ⎝ ⎠ ⎝ x ⎠ j= 0⎝ j ⎠ occurs when: 24 − 3 j = 0 24 = 3 j j = 8 3 3 2 24−3j ⎛12⎞ 12! 12⋅11⋅10 ⋅9 The coefficient is ⎜ 495 8 ⎟ = = = ⎝ ⎠ 8!4! 4321 ⋅ ⋅ ⋅ 0 40. The x term in ∑ 9 j 9 ⎛9⎞ − ⎛ 1 ⎞ ⎛9⎞ ⎜ ⎟ x 1 2 j= 0 j ⎜− ⎟ = ∑ ⎜ ⎟ − ⎝ ⎠ ⎝ x ⎠ j= 0⎝ j⎠ occurs when: 9− 3j = 0 9= 3j j = 3 The coefficient is ⎛9⎞ ( ) 3 9! 9⋅8⋅7 ⎜ ⎟ − 1 =− =− =−84 ⎝3⎠ 3!6! 3⋅2⋅1 ( ) ( ) 9 j j 9−3j x 1271 © 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: Section 12.5: The Binomial Theorem
Page 39 and 40: Chapter 12 Review Exercises 50. 8 1
Page 41 and 42: Chapter 12 Review Exercises 29. 7
Page 43 and 44: Chapter 12 Review Exercises 51. I:
Page 45 and 46: Chapter 12 Review Exercises c. If t
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

Chapter 12 Sequences; Induction; the Binomial Theorem

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