# R_7 - Chapter R Review R.7 1 3 5 Rational Expressions...

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Unformatted text preview: Chapter R Review R.7 1. 3. 5. Rational Expressions lowest terms True 3x + 9 3(x + 3) 3 = (x - 3)(x + 3) = x - 3 2 x -9 x 2 - 2x x(x - 2 ) x 3x - 6 = 3(x - 2) = 3 24x 2 24x 2 4x = 6x(2x - 1) = 2x - 1 2 12x - 6x 2. 4. 6. Least Common Multiple False 4x 2 + 8x 4x(x + 2) x 12x + 24 = 12(x + 2) = 3 15x 2 + 24x 3x(5x + 8) 5x + 8 = = x 3x 2 3x 2 x 2 + 4x + 4 (x + 2)2 = (x - 4)(x + 4) x 2 - 16 7. 8. 9. 10. 11. y 2 - 25 ( y + 5)( y - 5) = ( y + 5)( y - 5) = y + 5 = 2 2y - 8y -10 2( y 2 - 4 y - 5) 2( y - 5)( y + 1) 2( y + 1) 3y 2 - y - 2 ( 3y + 2)( y -1) y -1 = = 3y 2 + 5y + 2 ( 3y + 2)( y + 1) y + 1 x -x2 - x(x -1) -x x = = =- 2 x + x - 2 (x + 2)(x -1) x + 2 x+ 2 x 2 + 5x -14 ( x + 7)( x - 2) ( x + 7)( x - 2) = = = -( x + 7) = -x - 7 2- x (-1)(-2 + x ) (-1)( x - 2) 2x 2 + 5x - 3 (2x -1)(x + 3) = = -(x + 3) = -x - 3 1- 2x (-1)(2x -1) 3x + 6 x 3(x + 2) x 3 2 = (x - 2)(x + 2 ) = 5x(x - 2) 2 2 5x x -4 5x 3 x2 3 x2 3x = = 2x 6x + 10 2x 2(3x + 5) 4(3 x + 5) 4x 2 x - 4 4x 2 x-4 2x 2 x = (x - 4)(x + 4) 2x = x + 4 x 2 - 16 38 13. x 2 + 4 x - 5 (x + 5)(x -1) x + 5 = = x 2 - 2x + 1 (x -1)2 x -1 12. 14. 15. 16. 17. 18. 19. Section R.7 20. 12 x2 - 1 12 (x - 1)(x + 1) 6(x + 1) 4x - 2 = x(x - 1) 2(2x - 1) = x(2x - 1) 2 x -x 4 ( x - 2) 4x - 8 12 12 8 = = -3x 12 - 6x -3x (-6)( x - 2) 3x 6x -27 2 3 (2x - 9) 2 3 = = 5x 4 x -18 5x 2(2x - 9) 5x x 2 - 3x - 10 x 2 + 4x - 21 (x - 5)( x + 2) ( x + 7)( x - 3) x - 3 = = x 2 + 2x - 35 x 2 + 9x +14 ( x + 7)( x - 5) ( x + 7)( x + 2) x + 7 Rational Expressions 21. 22. 23. 24. x2+ x- 6 x 2 - 25 (x - 2)( x + 3) ( x + 5)( x - 5) (x - 2)( x + 3)( x - 5) 2 = = 2 x + 4x -5 x + 2x -15 ( x + 5)( x -1) ( x + 5)( x - 3) ( x + 5)( x - 1)( x - 3) 6x 6x 2(x + 2) 4x x - 4 = 6x 2x + 4 = = 3x - 9 x 2 - 4 3x - 9 (x - 2)(x + 2) 3(x - 3) (x - 2)(x - 3) 2x + 4 2 25. 26. 12x 2 5x + 20 = 12x x -16 = 12x (x + 4)( x - 4) = 3(x - 4) 4x2 5x + 20 4 x 2 5(x + 4) 4x2 5x 2 x -16 8x 8x x +1 4x x 2 -1 = 8x x + 1 = = 2 10 x -1 10 ( x -1)( x + 1) 10 5( x -1) x +1 x -2 x -2 12x x - 2 12x 3 4x = 2 = 2 2 = x - 4x + 4 4x x - 4x + 4 4 x ( x - 2) x -2 12x 4-x 2 2 2 4 + x = 4 - x x -16 = (-1)( x - 4 ) ( x + 4 )( x - 4 ) = -( x - 4 ) = - ( x - 4 ) 4x 4+ x 4x x+4 4x 4x 4x 2 x -16 3+ x 3 9x 3 9x 3 9x 3 3 - x = 3 + x 9x = x + 3 = =- 2 x 2 - 9 3 - x x 2 - 9 (-1)( x - 3) ( x + 3)( x - 3) -( x - 3) 2 ( x - 3) 9x 3 27. 28. 29. 30. 39 Chapter R Review 31. x 2 + 7x + 12 2 2 2 x 2 - 7x + 12 = x + 7x + 12 x - x -12 = (x + 3)(x + 4) (x - 4)( x + 3) = (x + 3) x 2 + x -12 x 2 - 7x + 12 x 2 + x -12 (x - 3)(x - 4) (x + 4)( x - 3) (x - 3)2 x 2 - x -12 x 2 + 7x + 6 2 2 x 2 + x - 6 = x + 7x + 6 x + 5x + 6 = (x + 6)(x + 1) (x + 2)(x + 3) x 2 + 5x - 6 x 2 + x - 6 x 2 + 5x - 6 (x + 3)(x - 2) (x + 6)(x -1) x 2 + 5x + 6 (x + 1)(x + 2 ) = (x - 2)(x - 1) 2x 2 - x - 28 2 2 3x 2 - x - 2 = 2x - x - 28 3x + 11x + 6 = (2x + 7)(x - 4) (3x + 2)(x + 3) 4 x 2 + 16x + 7 3x 2 - x - 2 4 x 2 + 16x + 7 (3x + 2)(x -1) (2x + 7)(2x + 1) 3x 2 + 11x + 6 (x - 4)(x + 3) = (x - 1)(2x + 1) 9x 2 + 3x - 2 2 2 12x 2 + 5x - 2 = 9x + 3x - 2 8x -10x - 3 = (3x + 2)(3x -1) (4 x + 1)(2x - 3) 9x 2 - 6x + 1 12x 2 + 5x - 2 9x 2 - 6x + 1 (3x + 2)(4 x -1) (3x -1)2 8x 2 -10x - 3 ( 4x + 1)(2x - 3) = ( 4x - 1)(3x - 1) x 5 x+5 + = 2 2 2 x2 4 x 2 - 4 (x + 2)( x - 2) - = = 2x - 3 2x - 3 2x - 3 2x - 3 2 3x 2 9 3x 2 -9 3 ( x - 3) - = = 2x -1 2x -1 2x -1 2x -1 32. 33. 34. 35. 36. 3 6 3- 6 -3 3 - = = =- x x x x x 37. 38. 39. 40. 41. 42. x + 1 2x - 3 x + 1+ 2x - 3 3x - 2 = x -3 x-3+ x-3 = x -3 2x - 5 x + 4 2x - 5 + x + 4 3x - 1 = 3x + 2 3x + 2 + 3x + 2 = 3x + 2 3x + 5 2x - 4 (3x + 5) - (2x - 4) 3x + 5 - 2x + 4 x+9 = = 2x - 1 2x - 1 - 2x - 1 = 2x - 1 2x - 1 5x - 4 x +1 (5x - 4) - (x + 1) 5x - 4 - x - 1 4x - 5 = = 3x + 4 3x + 4 - 3x + 4 = 3x + 4 3x + 4 40 Section R.7 43. 44. 45. 4 x 4 x 4-x x -2 + 2- x = x - 2 - x -2 = x -2 6 x 6 x x+6 x - 1 - 1- x = x - 1 + x - 1 = x - 1 Rational Expressions 4 2 4(x + 2 ) 2(x - 1) 4x + 8 - 2x + 2 2x + 10 x - 1 - x + 2 = (x - 1)(x + 2 ) - (x + 2)(x - 1) = (x + 2)(x - 1) = (x + 2)(x - 1) 2(x + 5) = (x + 2)(x - 1) 2 5 2(x - 5) 5(x + 5) 2x -10 - 5x -25 -3x - 35 - = - = = x + 5 x - 5 (x + 5)(x - 5) (x + 5)(x - 5) (x + 5)(x - 5) (x + 5)(x -5) 3x + 35 =- (x + 5)(x - 5) x 2x - 3 x(x - 1) (2x - 3)(x + 1) x 2 - x + 2x 2 - x - 3 + x - 1 = (x + 1)(x - 1) + (x - 1)(x + 1) = x +1 (x - 1)(x + 1) 3x 2 - 2x - 3 = (x - 1)(x + 1) 3x 2x 3x(x + 3) 2x(x - 4) 3x 2 + 9x + 2x 2 - 8x + x + 3 = (x - 4 ) (x + 3) + (x - 4)(x + 3) = x-4 (x - 4)(x + 3) 2 5x + x = (x - 4)( x + 3) x (5x + 1) = (x - 4)( x + 3) x - 3 x + 4 (x - 3)(x - 2) (x + 4 ) (x + 2) x 2 - 5x + 6 - (x 2 + 6x + 8) x + 2 - x - 2 = (x + 2)(x - 2 ) - (x - 2)(x + 2) = (x + 2)(x - 2) 2 2 x - 5x + 6 - x - 6x - 8 = (x + 2)(x - 2) -11x - 2 11x + 2 = =- (x + 2)(x - 2) ( x + 2)( x - 2) 2x - 3 2x + 1 ( 2x - 3)(x + 1) ( 2x + 1)(x - 1) 2x 2 - x - 3 - (2x 2 - x - 1) x - 1 - x + 1 = (x - 1)(x + 1) - (x + 1)(x - 1) = (x + 1)(x - 1) 2 2 2x - x - 3 - 2x + x + 1 -2 = = (x + 1)(x -1) (x + 1)(x -1) 2 =- (x + 1)(x -1) 2( x 2 - 2) x 1 x2 + x2- 4 2x 2 - 4 + = = = x 2 - 4 x ( x) ( x 2 - 4) ( x )( x2 - 4) ( x )(x - 2)( x + 2) 41 46. 47. 48. 49. 50. 51. Chapter R 52. Review (x -1)( x 2 +1) + x 4 x 3 - x 2 + x -1+ x 4 x 4 + x 3 - x 2 + x -1 x -1 x + 2 = = = x3 x +1 (x 3 )( x 2 +1) ( x 3)( x 2 +1) ( x 3)( x 2 +1) x 2 - 4 = ( x + 2)( x -2) x 2 - x - 2 = ( x + 1)( x - 2) LCM is 53. ( x + 2)( x - 2)( x + 1) 55. 2 54. x 2 - x -12 = (x + 3)( x - 4 ) x 2 - 8x + 16 = ( x - 4 )( x - 4 ) LCM is x 3 - x = x ( x 2 -1) = x ( x +1)( x -1) x 2 - x = x ( x -1) LCM is x ( x + 1)( x -1) ( x + 3)( x - 4 ) 56. 2x 2 - x -15 = (2x + 5)( x - 3) 3x 2 - 27 = 3 ( x 2 - 9) = 3( x + 3)(x - 3) LCM is 3(2x + 5)( x - 3)( x + 3) 57. 2x 3 - x 2 = x 2 (2x -1) x3 = x3 4x 3 - 4x 2 + x = x (4 x 2 - 4x + 1) = x (2x - 1)(2x -1) LCM is x 3 (2x -1) 58. x -3= x -3 x 2 + 3x = x ( x + 3) 2 x 3 - 9x = x ( x 2 - 9) = x ( x + 3)( x - 3) LCM is x ( x + 3)( x - 3) 59. x 3 - x = x ( x 2 -1) = x ( x +1)( x -1) x 3 -1 = ( x -1)( x 2 + x + 1) x 3 - 2x 2 + x = x ( x 2 - 2x + 1) = x ( x -1)( x -1) LCM is x ( x + 1)( x -1) ( x 2 + x + 1) 2 60. x 2 + 4 x + 4 = ( x + 2)( x + 2) x 3 + 2x 2 = x 2 ( x + 2) ( x + 2) 3 = ( x + 2) 3 3 LCM is x 2 ( x + 2) 42 Section R.7 61. Rational Expressions x x x x - = - x 2 - 7x + 6 x 2 - 2x - 24 (x - 6)(x - 1) (x - 6)(x + 4) x(x + 4) x(x -1) = - (x - 6)(x -1)(x + 4) (x - 6)(x + 4)( x -1) x2 + 4x - x2 + x (x - 6)(x + 4)( x -1) 5x = (x - 6)(x + 4)( x -1) = 62. x x +1 x x +1 - 2 = - x - 3 x + 5x - 24 x - 3 (x - 3)(x + 8) x(x + 8) x +1 x 2 + 8x - x -1 = - = (x - 3)(x + 8) (x - 3)(x + 8) (x - 3)(x + 8) x 2 + 7x -1 = (x - 3)(x + 8) 63. 4x 2 4x 2 - = - x 2 - 4 x 2 + x - 6 (x - 2)(x + 2 ) (x + 3)(x - 2) 4 x(x + 3) 2(x + 2) = - (x - 2)(x + 2)(x + 3) (x + 3)(x - 2)(x + 2) = = 4 x 2 + 12x - 2x - 4 (x - 2)(x + 2)(x + 3) 4 x 2 + 10x - 4 (x - 2)(x + 2)(x + 3) 2(2x 2 + 5x - 2) = (x - 2)(x + 2)(x + 3) 64. 3x x-4 3x x-4 3x(x -1) x-4 3x 2 - 3x - x + 4 - 2 = - = - = x -1 x - 2x + 1 x -1 (x -1)2 (x -1)(x -1) (x -1)2 (x -1)2 3x 2 - 4x + 4 = (x - 1)2 3 + 3( x + 1) 2( x -1) 2 3x + 3 + 2x - 2 + = 2 = 2 2 2 2 ( x -1)( x + 1) ( x -1) ( x + 1)( x + 1) ( x -1)( x + 1) ( x -1) ( x -1) ( x + 1) 5x + 1 = 2 2 ( x -1) ( x +1) 65. ( x -1) ( x + 1) 2 43 Chapter R 66. 2 2 Review - 6 2 ( x + 2) ( x -1) ( x + 2)( x -1) ( x + 2) ( x -1)( x -1) ( x + 2)( x -1) ( x + 2) -2(2x + 7) 2x - 2 - 6x -12 -4 x -14 = 2 2 = 2 2 = 2 2 ( x + 2) ( x -1) ( x + 2) ( x -1) ( x + 2) ( x -1) 2(2x + 7) =- 2 2 ( x + 2) ( x -1) 2 2 = 2( x -1) - 6( x + 2) 67. x +4 2x +3 x +4 2x + 3 - = - x 2 - x - 2 x 2 + 2x - 8 (x - 2 ) (x + 1) (x + 4)(x - 2) (x + 4)( x + 4) (2x + 3)(x + 1) = - (x - 2)(x + 1)(x + 4) (x + 4)( x - 2)(x + 1) = = x 2 + 8x + 16 - (2x 2 + 5x + 3) (x - 2)(x + 1)(x + 4) -x 2 + 3x + 13 (x - 2)(x + 1)(x + 4) 68. 2x - 3 x-2 2x - 3 x-2 - 2 = (x + 1)(x + 7) - x + 8x + 7 (x + 1) (x + 1)2 (2x - 3)(x + 1) (x - 2)(x + 7) 2x 2 - x - 3 - (x 2 + 5x - 1 4 ) = - = (x + 1)(x + 7)(x + 1) (x + 1)2 (x + 7) (x + 1)2 (x + 7) 2 = x - 6x + 11 (x + 1)2 (x + 7) 2 69. 1 2 3 1 2 3 x ( x +1)( x -1) - 2x( x -1) + 3( x +1) - 2 + 3 - + 2 = 2 = x x + x x -x x x ( x +1) x ( x -1) x 2 ( x +1)( x -1) x ( x 2 -1) - 2x 2 + 2x + 3x + 3 x 3 - x - 2x 2 + 5x + 3 = = x 2 ( x + 1)( x -1) x 2 ( x + 1)( x -1) x 3 - 2x 2 + 4 x + 3 = 2 x ( x + 1)( x -1) 70. x 2 x +1 x 2 x +1 x 3 + 2x( x - 1) - ( x + 1)( x - 1) + - 3 = + - 2 = 2 2 2 2 x 2( x -1) ( x -1) x x - x ( x -1) x x ( x -1) 2 = x 3 + 2x ( x 2 - 2x + 1) - ( x 2 -1) x 2 ( x -1) 2 = x 3 + 2x 3 - 4 x 2 + 2x - x 2 + 1 2 x 2 ( x -1) 3x 3 - 5x 2 + 2x + 1 = 2 x 2 ( x -1) 71. 1 1 1 x + h - x = h 1 x x+h - = h (x + h)x x(x + h) 1 x - x - h -h 1 =- = h x(x + h) hx(x + h) x(x + h) 44 Section R.7 1 1 1 - 2= 2 h (x + h) x Rational Expressions 72. 1 x2 (x + h) 2 1 x 2 - (x 2 + 2xh + h 2 ) - 2 = h (x + h) 2 x 2 x (x + h) 2 h x 2 (x + h) 2 2 -2x h - h h(-2x - h) -2x - h = 2 = 2 = 2 2 2 hx (x + h) hx (x + h) x (x + h) 2 2x + h =- 2 x (x + h) 2 1+ 73. 1 x 1 x +1 + x = x x = x = x +1 x = x +1 1 x 1 x -1 x x -1 x -1 1- - x x x x 4+ 1 4x2 1 4x2 +1 + 2 2 2 4x2 +1 x2 = x2 x = x2 = 4x +1 x = 2 1 3x 2 1 3x 2 -1 x2 3x 2 -1 3x -1 3- 2 - x x2 x2 x2 74. 2- 75. x +1 2x x + 1 x -1 - x -1 x + 1 ( x -1)( x + 1) x = x x = x = = x -1 3x + 3 x -1 4 x + 2 x 4x + 2 2x(2x + 1) 3+ + x +1 x +1 x +1 x +1 1- 76. x x +1 x 1 - x x +1 = x +1 x +1 = x +1 = 1 x = x -1 2x x -1 x + 1 x + 1 x + 1 (x + 1)2 2- - x x x x 2 2 x + 4 x - 3 (x + 4)( x + 1) - (x - 3)(x - 2) x + 5x + 4 - (x - 5x + 6) - (x - 2)(x + 1) x - 2 x + 1 = (x - 2)(x + 1) (x + 1)(x - 2) = x +1 x +1 x +1 10x - 2 1 = (x - 2)(x + 1) x + 1 2(5x -1) = (x - 2)(x + 1)2 77. 78. (x - 2)(x - 2) x(x + 1) x 2 - 4 x + 4 - (x 2 + x) x -2 x - - (x - 2)(x + 1) x + 1 x - 2 = (x + 1)(x - 2) (x - 2)(x + 1) = x+3 x+3 x+3 -5x + 4 1 = (x - 2)(x + 1) x + 3 -5x + 4 = (x - 2)(x + 1)(x + 3) 45 Chapter R Review 79. 2 2 x - 2 x -1 (x - 2)(x + 1) + (x -1)(x + 2) x - x - 2 + x + x - 2 + (x + 2)(x + 1) x + 2 x + 1 = (x + 2)(x + 1) (x + 1)(x + 2) = x 2x - 3 x2 (2x - 3)(x + 1) x 2 - (2x 2 - x - 3) - - x +1 x (x + 1)(x) x(x + 1) x(x + 1) 2 2x - 4 2(x 2 - 2) x(x + 1) 2x(x 2 - 2) (x + 2)(x + 1) = = = -x 2 + x + 3 (x + 2)(x + 1) -(x 2 - x - 3) -(x + 2)(x 2 - x - 3) x(x + 1) = -2x(x 2 - 2) (x + 2)(x 2 - x - 3) 80. (2x + 5)(x - 3) x2 2x 2 - x -15 - x 2 2x + 5 x - - x(x - 3) x(x - 3) x(x - 3) x x-3 = 2 2 2 2 = 3 2 x (x + 1) x (x + 3) (x - 3)(x + 1) x + 3x - (x 3 - x 2 - 5x - 3) - - x-3 x+3 (x - 3)(x + 3) (x - 3)(x + 3) (x - 3)(x + 3) 2 x - x -15 x 2 - x -15 (x - 3)(x + 3) x(x - 3) = = 4 x 2 + 5x + 3 x(x - 3) 4 x 2 + 5x + 3 (x - 3)(x + 3) = 1 1- 1 x 1 1- 1 1- x = 1- (x 2 - x -15)(x + 3) x(4 x 2 + 5x + 3) 81. 1- 1 x x -1- x -1 1 = 1-1 = = =- x -1 x -1 x -1 x -1 x -1 x 1 1 1- x 1- x x + 1- x 1 = 1- = 1-1 = 1+ 1 = = 1- x -1 -x -x x x x 1- x 1- x 82. 1- = 1- 83. (2x + 3) 3 - ( 3x - 5) 2 = 6x + 9 - 6x + 10 = 19 2 2 2 (3x - 5) (3x - 5) (3x - 5) (4 x + 1) 5 - (5x - 2) 4 = 20x + 5 - 20x + 8 = 13 2 2 2 (5x - 2) (5x - 2) (5x - 2) x 2x - ( x 2 + 1) 1 84. 85. (x (x 2 + 1) 2 = 2x 2 - x 2 -1 (x 2 + 1) 2 = (x x 2 -1 2 + 1) 2 = ( x + 1)( x -1) (x 2 + 1) 2 86. x 2x - ( x 2 - 4 ) 1 2 - 4) 2 = 2x 2 - x 2 + 4 (x 2 - 4) 2 = (x x2 + 4 2 - 4) 2 = x2 + 4 2 2 ( x + 2) ( x - 2) 46 Section R.7 Rational Expressions 87. (3x + 1) 2x - x 2 3 = 6x 2 + 2x - 3x 2 = 3x 2 + 2x = x (3x + 2) 2 2 2 2 ( 3x + 1) (3x + 1) ( 3x + 1) (3x + 1) (2x - 5) 3x 2 - x 3 2 = 6x 3 -15x 2 - 2x 3 = 4 x 3 -15x 2 = x 2 (4 x -15) 2 2 2 2 (2x - 5) (2x - 5) (2x - 5) (2x - 5) 88. 89. (x 2 + 1) 3 - ( 3x + 4 ) 2x (x 2 + 1) 2 = = 3x 2 + 3 - 6x 2 - 8x -( 3x 2 + 8x - 3) (x 2 2 + 1) 2 2 = -3x 2 - 8x + 3 (x 2 + 1) 2 =- (3x -1)( x + 3) (x + 1) (x 2 + 1) 2 90. (x 2 + 9) 2 - (2x - 5) 2x (x 2 + 9) 2 = 2x 2 + 18 - 4 x 2 + 10x (x 2 + 9) 2 = -2x 2 + 10x + 18 (x 2 + 9) 2 = -2( x 2 - 5x - 9) (x 2 + 9) 2 91. 1 1 1 = (n -1) + f R1 R2 1 R1 R2 1 R R = (n -1) + R1 R2 1 2 = (n -1)( R2 + R1 ) f f R1 R2 f 1 R1 R2 = f = R1 R2 (n -1)( R2 + R1 ) (n -1)( R2 + R1 ) f = 0.1(0.2) 0.02 0.02 2 = = = meter (1.5 -1)(0.2 + 0.1) 0.5(0.3) 0.15 15 92. R R + R1 R3 + R R 1 1 1 1 = R + R +R = 2 3 RR R 1 2 R 1 2 3 1 2 3 R1R2R3 5 4 10 200 20 R= = = = ohms R2 R3 + R1 R3 + R1R2 4 10 + 510 + 5 4 110 11 1+ 1 x +1 = a = 1, b = 1, c = 0 x x 1 1+ 1 x = 1+ 1 x x + 1+ x 2x + 1 = 1+ = = a = 2, b = 1, c = 1 x +1 x +1 x +1 x +1 x 93. 1+ 47 Chapter R 1+ 1+ 1 1 1+ 1 1+ 1+ 1 1 1+ 1 x Review = 1+ 1 x + 1 2x + 1+ x + 1 3x + 2 = 1+ = = a = 3, b = 2, c = 1 2x + 1 2x + 1 2x + 1 2x + 1 x +1 1 2x + 1 3x + 2 + 2x + 1 5x + 3 = 1+ = = a = 5, b = 3, c = 2 3x + 2 3x + 2 3x + 2 3x + 2 2x + 1 1+ = 1+ 1 x If we continue this process, the values of a, b and c produce the following sequences: a :1,2,3,5,8,13,21,.... b :1,1,2,3,5,8,13,21,..... c :0,1,1,2,3,5,8,13,21,..... In the case of (b), we have the Fibonacci Sequence, where the next value in the list is obtained from the sum of the previous 2 values in the list. 9495. Answers will vary. 48 ...
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