Chapter 12 Sequences; Induction; the Binomial Theorem

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Chapter 12: Sequences; Induction; the Binomial Theorem 28. n = 1: 2 1 − 1+ 41 = 41 is a prime number. n = 41: 2 2 41 − 41+ 41 = 41 is not a prime number. 29. II: If 2+ 4+ 6+ + 2k = k + k + 2, then 2+ 4+ 6+ + 2k+ 2( k+ 1) = 2+ 4+ 6+ + 2k + 2k+ 2 I: 30. I: [ ] 2 = k + k+ 2+ 2k+ 2 2 = ( k + 2k+ 1) + ( k+ 1) + 2 2 = ( k + 1) + ( k+ 1) + 2 2 n = 1: 2⋅ 1= 2 and1 + 1+ 2 = 4 ≠ 2 1 11 − ⎛1− r ⎞ n= 1: ar = a and a ⎜ = a 1− r ⎟ ⎝ ⎠ k 2 k −1 ⎛1− r ⎞ II: If a+ ar+ ar + + ar = a ⎜ 1− r ⎟ , ⎝ ⎠ then 2 k− 1 k+ 1−1 a+ ar+ ar + + ar + ar 2 k−1 k = ⎡a ar ar ar ⎤ ⎣ + + + + ⎦ + ar ⎛ k k k 1 −r ⎞ k a(1 − r ) + ar (1 −r) = a ⎜ ar 1 r ⎟ + = ⎝ − ⎠ 1−r k k k+ 1 k+ 1 a− ar + ar −ar ⎛1−r ⎞ = = a 1−r ⎜ 1−r ⎟ ⎝ ⎠ Conditions I and II are satisfied; the statement is true. 31. I: n = 1: 1(1 −1) a+ (1− 1) d = a and 1⋅ a+ d = a 2 II: If a+ ( a+ d) + ( a+ 2 d) + + [ a+ ( k−1) d] kk ( −1) = ka + d 2 then a+ ( a+ d) + ( a+ 2 d) + + [ a+ ( k − 1) d] + ( a+ kd) = [ a+ ( a+ d) + ( a+ 2 d) + + [ a+ ( k − 1) d] ] + ( a+ kd) kk ( −1) = ka + d + ( a + kd) 2 ⎡kk ( −1) ⎤ = ( k+ 1) a+ d ⎢ + k 2 ⎥ ⎣ ⎦ 2 ⎡ 2 k − k + 2k⎤ = ( k+ 1) a+ d ⎢⎣ 2 ⎥⎦ ⎡ 2 k + k⎤ = ( k+ 1) a+ d ⎢⎣ 2 ⎥⎦ ⎡( k + 1) k⎤ = ( k+ 1) a+ d⎢ 2 ⎥ ⎣ ⎦ ⎡( k + 1) ⎡ ( k + 1) −1⎤⎤ = ( k + 1) a+ d ⎣ ⎦ ⎢ ⎥ ⎢⎣ 2 ⎥⎦ Conditions I and II are satisfied; the statement is true. 32. I: n = 4: The number of diagonals of a quadrilateral is 1 4(4 3) 2 2 ⋅ − = . II: Assume that for any integer k, the number of diagonals of a convex polygon with k sides (k vertices) is 1 ( 3) 2 ⋅ kk− . A convex polygon with k + 1 sides ( k + 1 vertices) consists of a convex polygon with k sides (k vertices) plus a triangle, for a total of ( k + 1) vertices. The diagonals of this k + 1-sided convex polygon consist of the diagonals of the k-sided polygon plus k − 1 additional diagonals. For example, consider the following diagrams. k + 1 = 6 sides k = 5 sides k − 1 = 4 new diagonals Thus, we have the equation: 1 1 2 3 kk ( − 3) + ( k− 1) = k − k+ k−1 2 2 2 1 2 1 = k − k−1 2 2 1 2 = ( k −k−2) 2 1 = ( k + 1)( k− 2) 2 1 = ( k + 1)[( k+ 1) − 3] 2 Conditions I and II are satisfied; the statement is true. 1268 © 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 33. I: n = 3: (3 −2) ⋅ 180°= 180 ° which is the sum of the angles of a triangle. II: Assume that for any integer k, the sum of the angles of a convex polygon with k sides is ( k −2) ⋅ 180°. A convex polygon with k + 1 sides consists of a convex polygon with k sides plus a triangle. Thus, the sum of the angles is ( k −2) ⋅ 180°+ 180 °= [( k+ 1) −2] ⋅ 180 ° . Conditions I and II are satisfied; the statement is true. 34. Answers will vary. Section 12.5 1. Pascal Triangle 2. ⎛6⎞ 6! 654321 ⋅ ⋅ ⋅ ⋅ ⋅ 65 ⋅ ⎜ ⎟= = = = 15 ⎝2⎠ 2!4! 2⋅1⋅4⋅3⋅2⋅1 2⋅1 ⎛n⎞ n! 3. False; ⎜ ⎟ = ⎝ j ⎠ j! ( n− j) ! 4. Binomial Theorem 5. 6. ⎛5⎞ 5! 54321 ⋅ ⋅ ⋅ ⋅ 54 ⋅ ⎜ ⎟= = = = 10 ⎝3⎠ 3! 2! 3⋅2⋅1⋅2⋅1 2⋅1 ⎛7⎞ 7! 7654321 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 765 ⋅ ⋅ ⎜ ⎟= = = = 35 ⎝3⎠ 3!4! 3214321 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 321 ⋅ ⋅ 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. ⎛7⎞ 7! 7⋅6⋅5⋅4⋅3⋅2⋅1 7⋅6 ⎜ ⎟= = = = 21 ⎝5⎠ 5!2! 5432121 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 21 ⋅ ⎛9⎞ 9! 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 9⋅8 ⎜ ⎟= = = = 36 ⎝7⎠ 7!2! 765432121 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 21 ⋅ ⎛50⎞ 50! 50⋅ 49! 50 ⎜ ⎟= = = = 50 ⎝49⎠ 49!1! 49! ⋅1 1 ⎛100⎞ 100! 100⋅99⋅98! 100⋅99 ⎜ ⎟= = = = 4950 ⎝ 98 ⎠ 98!2! 98!21 ⋅ ⋅ 21 ⋅ ⎛1000⎞ 1000! 1 ⎜ ⎟ = = = 1 ⎝1000⎠ 1000!0! 1 ⎛1000⎞ 1000! 1 ⎜ ⎟ = = = 1 ⎝ 0 ⎠ 0!1000! 1 ⎛55⎞ 55! ⎜ ⎟ = ≈ 1.8664 × 10 ⎝23⎠ 23!32! ⎛60⎞ 60! ⎜ ⎟ = ≈ 4.1918 × 10 ⎝20⎠ 20! 40! ⎛47⎞ 47! ⎜ ⎟ = ≈ 1.4834 × 10 ⎝25⎠ 25! 22! ⎛37⎞ 37! ⎜ ⎟ = ≈ 1.7673 × 10 ⎝19 ⎠ 19!18! 15 15 13 10 _________________________________________________________________________________________________ 17. 18. 19. ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ( x + 1) = ⎜ ⎟x + ⎜ ⎟x + ⎜ ⎟x + ⎜ ⎟x + ⎜ ⎟x + ⎜ ⎟x ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 5 5 4 3 2 1 0 5 4 3 2 = x + 5x + 10x + 10x + 5x+ 1 ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ⎛5⎞ ( x − 1) = ⎜ ⎟x + ⎜ ⎟( − 1) x + ⎜ ⎟( − 1) x + ⎜ ⎟( − 1) x + ⎜ ⎟( − 1) x + ⎜ ⎟( −1) x ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ 5 5 4 2 3 3 2 4 1 5 0 5 4 3 2 = x − 5x + 10x − 10x + 5x−1 ⎛6⎞ ⎛6⎞ ⎛6⎞ ⎛6⎞ ⎛6⎞ ⎛6⎞ ⎛6⎞ ( x− 2) = ⎜ ⎟x + ⎜ ⎟x ( − 2) + ⎜ ⎟x ( − 2) + ⎜ ⎟x ( − 2) + ⎜ ⎟x ( − 2) + ⎜ ⎟x( − 2) + ⎜ ⎟x ( −2) ⎝0⎠ ⎝1⎠ ⎝2⎠ ⎝3⎠ ⎝4⎠ ⎝5⎠ ⎝6⎠ 6 6 5 4 2 3 3 2 4 5 0 6 6 5 4 3 2 = x + 6 x ( − 2) + 15x ⋅ 4 + 20 x ( − 8) + 15x ⋅ 16 + 6 x⋅( − 32) + 64 6 5 4 3 2 = x − 12x + 60x − 160x + 240x − 192x+ 64 1269 © 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 31 and 32: Section 12.4: Mathematical Inductio
Page 33: Section 12.4: Mathematical Inductio
Page 37 and 38: 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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