71. For the, arithmetic sequence, (an) whose second and fourth terms are, 72 and 712 respectively, nd a) (13 1)) Ss- 72. Use, concepts of sequences to nd the, sum of the first 100 even integers: i.e., 2 + 4 + (i + + 200. 73. Use, the concepts of gemnetric sequences to convert (1.57 to a rational number in lmvest terms. Cr'euted/Revised the [Math Department Math 2412 , Final Exam Review Problems . . . . . , 13 74. Evaluate and simplify the following binonnal cuel'hcelent: (_1 ) 75. Use the Binomial Theorem to expand (I 2) l. 1 1 2 -l 8 76. Find the exact value of the sum of the rst 20 terms of the sequence: 77777 . . . .. (i- 3' 3- 3 i 3 77. Find the fourth term of (21 - y)7. 78. Use the Principle of Mathematical Induction to prove 4+8+...+_]n =2n(n +1) for each n E N. (a) Identify 5(11): (h) Base Case: Show that 5(1) is true. (c) State the Inductive Hypothesis: Assume that 50:) is true: (d) Identify S(k+ 1): (e) Inductive Step: Use the Inductive Hypothesis tn shmv that EU; + 1) is true. 79. Use the Principle of Mathematical Inductiun to prove 3+ 11 + + (8n -5) = "(411 1) for each n E N. (a) Identify 5(11): (h) Base Case: Show that 5(1) is true. (c) State the Inductive Hypothesis: Assume that 50:) is true: (d) Identify S(k+ 1): (e) Inductive Step: Use the Inductive Hypothesis to show that EU; + 1) is true m 2 n4 80. Find the exact value of the following sum: 2 15 (g) . [:1 - 81. Use the concepts of geometric sequences to write 1.23456 as a rational number in simplest form. 3(2)\"l . if n is odd 82. Let a,1 be the sequence defined by a" = { 10 + 301 _ 1) . if n is even , Find the sun]: Li] + (12 + + in- 7 Graded/Revised the Illath Department Math 3412 - Final EJ'IITH Itcluru' Problems 83. Use properties of limits to lint] the indicated limit. ling\"; 7 8) cw. Select the correet choice helmv and fill in any answer lilanks in your choice. 0 A. lllltfll' 7 8) : 4.) O B. The limit does not exist.