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# quiz1_solns - MATH 2602 — Quiz 7 1 — N ame Prove both...

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Unformatted text preview: MATH 2602 — Quiz 7% 1 — May 13, 2009 N ame: Prove both of the following pr0positions using mathematical induction. It is up to you whether to use ordinary induction or strong induction. Both problems are worth 10 points. Show all of your work, and try to be as neat as possible. Points may be deducted if solutions are unclear. Use the back of the paper if you need more Space. 'H. Proposition 1. For every natural number n 2 O, we haue Z23- : 2"+1 — 1. i=0 -“ . t r“ 1‘: Q ‘ ' ﬁﬂgﬁ Cage“ H a ; EOQL: gmx‘Lre‘ga a: \f/ . 7 ,3 ,I t. q] ‘Y‘Cbc‘imiﬁ: {riff ‘ :1 {Sﬂ‘uu‘ er ma‘f \ W“ ‘ h“ n ‘ ﬂ 9 L t+\ v .9934; ‘I m 22 :22)“.— g; i wig»: ii 2" {J t 3, 1* in: 0 Eu? 0 3 R c .i ' .1... ﬂ . mi Am": r, . ’ a: w _ .3; W "L 2 1! \$3; ~wa figuéina‘ 2( 2 W H 4L 3 i i {b0 x Q ‘ : jihad)“ \ V Proposition 2. Consider the recursively deﬁned sequence ahag, where (11 z 1, a2 2 ...... 3, a3 = 6 and an = (1,1,1 + and + (111-3 for all n 2 4. Then for every natural number n 2 1, we have an < 2”. (That is, you want to prove that an is STRICTLY less than 2”. Hint: be careful with, your number of base eases.) ""‘ 1:5“ 5 § _ r “we I” 2, W gh“.mr§_,;,'f:— gage Gamay» CM: ‘42 “A: V: “aﬁﬁ‘ﬂ’hl {3W {3’ {4‘ “" qH : q? AF 65-; 12;; é’g' 2': 1::- i: ‘/ wax—t n -'*~. ., v. ' = a. x 61' MATH 2602 — Quiz 4% 1 _ May 13, 2009 N alne: Prove both of the following propositions using mathematical induction. It is up to you whether to use ordinary induction or strong induction. Both problems are worth 10 points. Show all of your work, and try to be as neat as possible. Points may be deducted if solutions are unclear. Use the back of the paper if you need more Space. Tl. Proposition 1. For every natural number n 2 O, we have 2 2‘- 71:0 {:0 — 2"+1—1. “(f ca-‘H ' m- is: a “,1 v” I“ iﬂciocnli‘ec Erie". ‘, Ragnar-2e “l v41”- Hf my ' mx . ~ , n ; W ~ m 2" Z Elsi 55' EZﬂ «H r» KW? M M i=0 .5 . Ii L L: 0 {17-0 in mi ,1 \ imu'ilri- l at; t \ ; ma" _ - '( 2 -\ 'i' ‘3' “29% {:0 ' ' b w \ v' Proposition 2. Consider the recursively deﬁned sequence a1,a2, . . . where a1 2 1, a2 2 3, a3 = 6 and an = anal + an_2 + an_3 for all n 2 4. Then for euery natural number n 2 1, we have an < 2". {That is, you want to prove that an is STRICTLY less than 2". Hint: be careful with your number of base cases.) _ i r ‘- 2 “a: 7 mixer”; liege Casey. an: \LZma v: 112,15. up; {01% is a w. . r a Marti ql’i : Q2 Jr Gig (n’i' ‘5‘3 3 3:" lag; Ll. / z A Wide We: a \$3." .lr'll' E‘!‘ fi‘ ,pmls n: Oink, him i. klﬁwa E Lara"; ( q t... a. .- — i... a 333 ll Gin-72, mi" i’iﬁw33Jr qh_1 «kahul»: J,C\hﬂ-%E"Axhp'2" “on? A 2%"! ﬂame 2 an x»- Wa m * 42f“ 2’” M . M j Mﬂ \w/ 35m \$8 - 93 “w H - :3 5 8% 2 95m“ 39\$ to; 0m 35 mozoibm Eovoﬂﬂodm mmgm 3333333 mdmgaos. Ha Mm Eu .8 VB: 273:2 no :mm 3%me Eaznao: 2, 385m Emccaou. we; EoEmBm Ed 201.3 5 @053 wroé m: cm VSE 29.? 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quiz1_solns - MATH 2602 — Quiz 7 1 — N ame Prove both...

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