ninth 2412 - Final Exam Review Problems 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. Identify 3.1'2 21'y + 9-2 + 21' + 2y 8 = [1 without applying a rotation of axes. State the center of (1+3)2 + (y 12)2 =16. Find a set of parametric equations for the rectangular equation (1' 3)2 = 5y. m2 (y') 9 2-5 "I After rotation of % radians. the equation of a conic C is = 1. Sketch the original conic C. Include the .r'yl-coordjnate axes. Find the coordinates of the foci {in the Iyplane). (i -l + 2 sin(l9) I a) State the eccentricity p b). Identify the conic. Consider the polar graph 1' = Rewrite Iy = 4 in a rotated I'y's_\\'ste1n without an r'y' term. . . r = sin 2t . . . _ Gwen ( ) . ehnunate the parameter to Obtain the rectangular equation. 9 = 3eos(t) Determine Whether each statement is TRUE (T) or FALSE (F). T F The sequence 2. 6. 21. 120. . .. is a geometric sequence. T F The sequence on = 5n + 12 is an arithmetic sequence. T F A sequence is a function from the natural numbers to the reals. . _ 1 T F For the geomtrlc sequence an = 3(2)"\"l. r is 2' T F 2+4+8+16+...=. 1 2 .- .' 10 T F 10+5++3+,,.= 1' 1 _ _ 2 T F If (11 = (i and a5 = 2 in an arithmetic sequence. then {13 = 2. Find an expression for the 13m term [i.e.. general term) of the sequence 2. 1.4.7. 10. . . . . l5! Evaluate and simplify: . 2! 13! + 2 ! Let n E N. Evaluate and simplify: Q. 13. 5 (1)"(i 1)! 2111 ' Find the suln: i=1 For the arithmetic sequence (on) whose second and fourth terms are 2 and 12 respectively. nd a) (13 b) 58- Use concepts of sequences to nd the sum of the rst 100 even integers: i.e.. 2 + .1 + 6 + + 200. Use the concepts of geometric sequences to convert 0.57 to a rational number in lowest terms. Math 2412 - Final Exam Review Problems 74. Evaluate and simplify the following binomial coefficient: (). 75. Use the Binomial Theorem to expand (x - 2)4. 76. Find the exact value of the sum of the first 20 terms of the sequence: 6, 3' 3' 3' 3"." 1 124 8 77. Find the fourth term of (2x - y)?. 78. Use the Principle of Mathematical Induction to prove 4 +8 + ... + 4n = 2n(n +1) for each n E N. (a) Identify S(n): (b) Base Case: Show that S(1) is true. (c) State the Inductive Hypothesis: Assume that S(k) is true: (d) Identify S(k + 1): (e) Inductive Step: Use the Inductive Hypothesis to show that S(k + 1) is true. 79. Use the Principle of Mathematical Induction to prove 3 +11 + ...+ (8n -5) = n(4n -1) for each n E N. (a) Identify S(n): (b) Base Case: Show that S(1) is true. (c) State the Inductive Hypothesis: Assume that S(k) is true: (d) Identify S(k + 1): (e) Inductive Step: Use the Inductive Hypothesis to show that S(k + 1) is true. 80. Find the exact value of the following sum: E 15 (2)"-. 81. Use the concepts of geometric sequences to write 1.23456 as a rational number in simplest form. 82. Let an be the sequence defined by an = 3(2)"-1 , if n is odd 10 + 3(n - 1) , if n is even Find the sum: aj + a2 + ... + @10. 7 Math