CHAPTER 9 Sequences, Series, and Probability

1 CHAPTER 9 sequences , series , and ProbabilitySection and series ..819 Section Sequences and Partial Sums ..831 Section sequences and series ..840 Section Induction ..852 Section Binomial Theorem ..868 Section Principles ..877 Section ..882 Review Exercises ..888 Problem Solving ..898 Practice Test ..902 CHAPTER 9 sequences , series , and ProbabilitySection and Series819 Vocabulary ; upper; sumnth Given the general nth term in a sequence , you should be able to find, or list, some of the terms. You should be able to find an expression for the apparent nth term of a sequence . You should be able to use and evaluate factorials. You should be able to use summation notation for a sum.

2 You should know that the sum of the terms of a sequence is a 3 5 1 16a4 3 4 1 13a3 3 3 1 10a2 3 2 1 7a1 3 1 1 4an 3n 5 5 3 22a4 5 4 3 17a3 5 3 3 12a2 5 2 3 7a1 5 1 3 2an 5n 25 32a4 24 16a3 23 8a2 22 4a1 21 2an 12 5 132a4 12 4 116a3 12 3 18a2 12 2 14a1 12 1 12an 12 2 5 32a4 2 4 16a3 2 3 8a2 2 2 4a1 2 1 2an 2 12 5 132a4 12 4 116a3 12 3 18a2 12 2 14a1 12 1 12an 12 n820 CHAPTER 9 sequences , series , and 75a4 64 32a3 53a2 42 2a1 1 21 3an n 6 5 3 5 2 1 1537a4 6 4 3 4 2 1 2447a3 6 3 3 3 2 1 913a2 6 2 3 2 2 1 1211a1 6 1 3 1 2 1 3an 6n3n2 0a4 24 12a3 0a2 22 1a1 0an 1 1 55 2 57a4 44 2 23a3 33 2 35a2 22 2 12a1 11 2 13an nn 3 5 2 5 42 5 2 1 7451a4 3 4 2 4 42 4 2 1 1611a3 3 3 2 3 42 3 2 1 2819a2 3 2 2 2 42 2 2 1 149a1 3 1 2 1 42 1 2 1 2an 3n2 n 42n2 1 1 5 0a4 1 1 4 2a3 1 1 3 0a2 1 1 2 2a1 1 1 1 0an 1 1 2 1243 485243a4 2 181 16181a3 2 127 5327a2 2 19 179a1 2 13 53an 2 2535 32243a4 2434 1681a3 2333 827a2 2232 49a1 2131 23an 153 2a4 143 2 18a3 133 2a2 123 2a1 11 1an 1n3 2 Section and 103 52 103 25a4 103 42 103 16a3 103 32 103 9a2 103 22 103 4a1 101 10an 10n2 3 103 125a4 116a3 19a2 14a1 11

3 1an 1 1 555 1 56a4 1 444 1 45a3 1 333 1 34a2 1 222 1 23a1 1 111 1 12an 1 n nn 1 23a4 23a3 23a2 23a1 23an 5 4 3 60a4 4 3 2 24a3 3 2 1 6a2 2 1 0 0a1 1 0 1 0an n n 1 n 2 5 52 6 95a4 4 42 6 40a3 3 32 6 9a2 2 22 6 4a1 1 12 6 5an n n2 6 1 25 3 25 2 7324. a16 1 16 1 16 16 1 240 an 1 n 1 n n 1 4 11 2 11 2 3 4423926. a13 4 13 2 13 313 13 1 13 2 37130 an 4n2 n 3n n 1 n 2 3102an 2 4n29. 0 101018an 16 n 8 n 2nn n2n2 sequence graph (c).a10 811a1 4,an 8n 1822 CHAPTER 9 sequences , series , and graph (b).a1 4, a3 244 6an 8 as n an 8nn sequence graph (d).a10 1128a1 4,an 4 n graph (a).a1 4, a4 444! 25624 1023an 0 as n an 4nn! 1 n 1 3 3n 21, 4, 7, 10,13, .. , 7, 11, 15, 19.

4 Apparent pattern:Each term is one less than fourtimes n,which implies thatan 4n :Terms:1 3 2 7 311415519..n n2 10, 3, 8, 15, 24, .. pattern:Each term is the product of and twice n,which implies thatan 1 n 1 2n . 1 n 1n:Terms:122 4 3 6 4 8510..n an2, 4, 6, 8, 10, .. 1 n n 1n 2 23, 34, 45, 56, 67, .. pattern:Each term is divided by 2raised to the n,which implies that an 1 n 12n. 1 n 1 Terms: 12 14 18 116 .. an n: 1 2 3 4 ..n12, 14, 18, 116, .. n 12n 121, 33, 45, 57, 69, .. pattern:Each term is divided by 3 raised to the n,which implies that an 2n 1 Terms: 13 29 427 881 .. an n: 1 2 3 4 .. n13, 29, 427, 881, .. 1n21, 14, 19, 116, 125, .. pattern:Each term is the reciprocal of n!

5 , which implies that an 1n!. Terms: 1 12 16 124 1120 .. an n: 1 2 3 4 5 ..n1, 12, 16, 124, 1120, .. 1 n 11, 1, 1, 1, 1, ..Section and 1 1n1 11, 1 12, 1 13, 1 14, 1 15, .. pattern:Each term is divided by which implies that an 2n 1 n 1 !. n 1 !,2n 1 Terms: 1 2 222 236 2424 25120 .. an n: 1 2 3 4 5 6 .. n1, 2, 222, 236, 2424, 25120, .. :1234 5.. nTerms:.. Apparent pattern: Each term is the sum of 1 and the quantity 1 less than divided by which implies that an 1 2n ,2nan1 31321 15161 781 341 121 12, 1 34, 1 78, 1 1516, 1 3132, .. 12a4 12 4 2a4 12a3 12 8 4a3 12a2 12 16 8a2 12a1 12 32 16a1 32a1 32, ak 1 344! 8124 278a3 333! 276 92a2 322! 92a1 311! 3a0 300!

6 1an 3nn! a4 4 16 4 12a4 a3 4 20 4 16a3 a2 4 24 4 20a2 a1 4 28 4 24a1 28a1 28 and ak 1 ak a4 3 24 3 27a4 a3 3 21 3 24a3 a2 3 18 3 21a2 a1 3 15 3 18a1 15a1 15, ak 1 ak 2 a4 1 2 10 1 18a4 2 a3 1 2 6 1 10a3 2 a2 1 2 4 1 6a2 2 a1 1 2 3 1 4a1 3a1 3 and ak 1 2 ak 1 general,an 2n a4 2 12 2 14a4 a3 2 10 2 12a3 a2 2 8 2 10a2 a1 2 6 2 8a1 6a1 6 and ak 1 ak general,an 30 a4 5 10 5 5a4 a3 5 15 5 10a3 a2 5 20 5 15a2 a1 5 25 5 20a1 25a1 25, ak 1 ak general,an 81 13 n 1 81 3 13 n 13a4 13 3 1a4 13a3 13 9 3a3 13a2 13 27 9a2 13a1 13 81 27a1 81a1 81 and ak 1 general,an 14 2 n 2 a4 2 112 224a4 2 a3 2 56 112a3 2 a2 2 28 56a2 2 a1 2 14 28a1 14a1 14, ak 1 2 ak824 CHAPTER 9 sequences , series , and 15! 1120a3 14!

7 124a2 13! 16a1 12! 12a0 11! 1an 1 n 1 ! 42 4 1 ! 165 4 3 2 1 215a3 32 3 1 ! 94 3 2 1 38a2 22 2 1 ! 43 2 1 23a1 12 1 1 ! 12 1 12a0 02 0 1 ! 01 0an n2 n 1 ! 18! 140,320a3 16! 1720a2 14! 124a1 12! 12a0 10! 1an 1 2n 2n ! 1 2n ! 1 2 4 1 2 4 1 ! 1 99! 1362,880 1362,880a3 1 2 3 1 2 3 1 ! 1 77! 15040 15040a2 1 2 2 1 2 2 1 ! 1 55! 1120 1120a1 1 2 1 1 2 1 1 ! 1 33! 16 16a0 1 2 0 1 2 0 1 ! 1 11! 11 1an 1 2n 1 2n 1 ! !6! 1 2 3 41 2 3 4 5 6 15 6 !8! 1 2 3 4 51 2 3 4 5 6 7 8 16 7 8 !8! 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 101 9068. 24 251 600 25!23! 1 2 3 .. 23 24 251 2 3 .. 2369. n 1 n 1 !n! 1 2 3 .. n n 1 1 2 3 .. n n 1170. n 1 n 2 n 2 !n! 1 2 3 .. n n 1 n 2 1 2 3 .. n71. 12n 2n 1 2n 1 ! 2n 1 ! 1 2 3 .. 2n 1 1 2 3.

8 2n 1 2n 2n 1 72. 3n 11 3n 1 3n 1 ! 3n ! 1 2 3 .. 3n 3n 1 1 2 3 .. 3n 73. 5i 1 2i 1 2 1 4 1 6 1 8 1 10 1 3574. 6i 1 3i 1 3 1 1 3 2 1 3 3 1 3 4 1 3 5 1 3 6 1 5775. 4k 110 10 10 10 10 4076. 5k 15 5 5 5 5 5 4!4 4 3 2 14 6a3 3!3 3 2 13 2a2 2!2 2 12 1a1 1!1 11 1a0 0!0 undefinedan n!nSection and Series82577. 4i 0i2 02 12 22 32 42 3079. 3k 0 1k2 1 11 11 1 14 1 19 1 9581. 5k 2 k 1 2 k 3 3 2 1 4 2 0 5 2 1 6 2 2 8883. 4i 1 2i 21 22 23 24 3078. 110 5i 02i2 2 02 2 12 2 22 2 32 2 42 2 52 80. 5j 3 1j2 3 132 3 142 3 152 3 12442982. 4i 1 i 1 2 i 1 3 0 2 2 3 1 2 3 3 2 2 4 3 3 2 5 3 23884. 11 4j 0 2 j 2 0 2 1 2 2 2 3 2 485. 6j 1 24 3j 8186. 10j 13j 1 4k 0 1 kk 1 476088. 4k 0 1 kk! 1 13 2 13 3 .. 13 9 9i 1 1 51 2 51 3.

9 51 15 15i 151 i91. 2 18 3 2 28 3 2 38 3 .. 2 88 3 8i 1 2 i8 3 92. 1 16 2 1 26 2 .. 1 66 2 6k 1 1 k6 2 9 27 81 243 729 6i 1 1 i 12 14 18 .. 1128 120 121 122 123 .. 127 7n 0 12 122 132 142 .. 1202 20i 1 1 i 38 716 1532 3164 5i 12i 12i 199. 4i 15 12 i 5 12 5 12 2 5 12 3 5 12 4 3 12 4 13 5 .. 110 12 10k 11k k 2 24 68 2416 12032 72064 6k 1k!2k100. 242243 5i 12 13 i 2 13 1 2 13 2 2 13 3 2 13 4 2 13 5101. 3n 14 12 n 4 12 4 12 2 4 12 3 32102. 5132 4n 18 14 n 8 14 1 8 14 2 8 14 3 8 14 4826 CHAPTER 9 sequences , series , and Probability103. i 16 110 i .. using a calculator, we haveThe terms approach zero as Thus, we conclude that k 17 110 k . 100k 17 110 k 79. 50k 17 110 k 10k 17 110 k (a)(b)A40 $11, $ $ $ $ $ $ $ $ 5000 1 n, n 1, 2, 3.

10 104. 19 .. k 1 110 k 110 1102 1103 1104 1105 ..106. 29 .. 2 .. 2 .. i 12 110 i 2 110 1102 1103 1104 .. 108.(a)(b)(c)A240 100 101 240 1 $99, 100 101 60 1 $ 100 101 6 1 $ 100 101 5 1 $ 100 101 4 1 $ 100 101 3 1 $ 100 101 2 1 $ 100 101 1 1 $ (a) Linear model:(c)The quadratic model is a better 182 YearnActualLinearQuadraticDataModelModel 1998831130330819999357363362200010419424 4202001114814844802002125485455442003136 08605611(b) Quadratic model:(d) For the year 2008 we have the following predictions:Linear model: 908 storesQuadratic model: 995 storesSince the quadratic model is a better fit, the predictednumber of stores in 2008 is and Series827110.