quiz1_solns - MATH 2602 — Quiz 7% 1 — May 13, 2009 N...

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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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Human mm 83%: 89.3 was, 3§3gi 0.x. 3% Emma; w - I; s N. , a fifivflk «Um/mm 09%? n5...“ th 3w (A 9wumhigw‘ WSW v i .5 _ . x of “ 9w “a. a; + 2,. mi? 4,, Sfi g N ,\ 5an wwwgmw my“ wwymw fifim .rafifl hfifl‘, Fwy? ,fifim if V ‘ J 3 _ .. \r... wl Q J; “3.3.Am dwinuwfiéw a”! fiw?$WU\/z .pjlw #0,.54U nu. fithwm.N£.53M 5m: “a.” fie if «N95; .7 M‘Qfizw .w M:Q6aw H 33 K: mfifi «w “flung man MATH 2602 — Quiz 7% 1— May 13, 2009 N ame: 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. 13 Proposition 1. For every natural number n 2 0, we have 2 2" = 2"+1 —- 1. i=0 @QSQ cage. O O} ‘3“ 2L : 2: 201.91 J Lin - , if: {34: {we 3 Fi'ifiuc‘ne 'liTmfi ‘g'bf V3 r WH _ fl I i 5 g‘ 'x fl?" i _. a mi -r e": we awe. Wu Weir-3 Z a — )rIZ (W03 “‘3 w a? 1 i=0 ' (to net {fine-13H a ‘0'“ . . “[2 “Ml—QM: 2.}; my: 2, win my; neural Proposition 2. Consider the recursively defined sequence a1,a2,. . . where a1 2 1, a2 = 3, a3 : 6 and an 2 and + an_2 + anmg 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 cases.) Bragg CCQQC’Q.‘ {Rt-:Eézzzixf ’3 g 2 3 Warren“ 13 2M"! «1 2.W \/ r \ l , (x no), iii-«91% gym? _ W? or simphygaimlfl‘ex a £1“) +2fl #2 w r21 *2 “+1 .r - w "5 {View $4 - ,_ 1-712” - w “"2. 4&2” v“ Kim 88 - @E #w H - 3% a. 88 2 Ede 3.95 we? Om :6 moHFOSEm UHGUOmeowm swim Bmfirmawflnfl wsacnamou. Ha mm EU no Es 2:95: Ho :3 3&st 395030: Ow mflozm 53530:. We? EoEmBm 98 5.03: S @033. . 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This note was uploaded on 11/08/2009 for the course MATH 2602 taught by Professor Costello during the Spring '09 term at Georgia Institute of Technology.

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quiz1_solns - MATH 2602 — Quiz 7% 1 — May 13, 2009 N...

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