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graded hw 6

# graded hw 6 - Due Monday 31 October 2011 Homework 6\jff(l...

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Unformatted text preview: Due Monday 31 October, 2011 Homework 6 \jff (l) Prove by the principle of mathematical induction that if m EN then for each 71 EN there exists '1 a 6220 such that n 27" S (n+a) 2m. “(2) Prove using the method of smallest counterexamples that for all a; e N 1+3+5+~--+(2m~1)=:c2. Let n 6220 and let a EN». Prove by the method of smallest counterexamples that 7‘ an+1_1 Zak = 1+a+a2+a3+---+a” = —— [i=0 a~1 n3 < (n+l)3. (5) Prove using the method of smallest counterexamples that for all n E N, 52’“1 + 1 is divisible by 6. Prove using the method of smallest counterexamples that for all n E N , \ n I Z *22’ . 42”” : 2212:2n+1__2_ i=1 2(Zn_l) )4/ 7) Let n E ZN). Prove that there exist 51, 52, 5k 6 Z20 such that si aé sj, i,j 6 {1,2, . . .,k} nd 4 V <xh wﬂmq 1i n=231+352+~-+25k. 251 0\ m wam % Oc‘rdaarzwau Mmﬁm _ ' _ 1M (9 1 L9: wnbgmﬁmmm’c Vnexijﬁezgc nzmiﬂnmgm , Mam. pm: 2'“£(1+a)2m7ms\\$m mm ash SD (M) 621+ , m «@351 Asgvmsu pm) 5.1m Whath - _ _. AMUQM W‘- Smmméﬂnukwmqkmn Emma « _+_C : pom): Cmle s(n_+l*.a12“") : wmﬁmeLXQ‘. (‘lmscmazm d3ng W2)“ = \n sum 1 1A Mum I +0 Lam rm <— ywm “Mmrh'pmmghSm ml“ 1392+ Cn*I)Lz‘“.)\$ (wwamm) WWW MSG-mas paw) Emu; m m Wmng 0L . Wmmncm mucum wt) my mm. Wax 7 ‘ re. WK ema efﬁgy , meﬂm - “36+. ﬂax-027x“ ,max ._ '_ mm? m 3mm}: comr mm: 34711052. Erqe M SthJa l*3‘5*.-J(Z“34> If 14”; Lei- S“ \$13£X \ “56+ Kilns—Q? «511 / T34 abgwﬂvnm s 3% CD ,_ 3mm 43e 8 - 11mm , 8, ms 6) SWAN umﬁam‘ %&\5_ m * J. 34%; «Amm- ﬁg’OMﬂySh ‘1 _y16"|3 awe—mg)" ‘ _ , , A Bu ’v‘ K‘m hbrst .+(ZH-I\ = K1. TQM its/1071., WHQN‘ZD‘5*.,.}'(7:(nufl3,_1\=(nof')? V V \0 “3+ch *(Zno-a) = “34%” I+ 3+2» "(2?!0'3) +(2m-l) : “3' ms madms) ‘31 ‘mm m m. mam akmm’; “9,13% s is W‘zmm + midygumx \_*5*35_*. +m—Q=x? «mm. ,, ‘8. mm“. _ \$3.:- 1t&*6‘*33* ‘ .331“ = a. Man, 3M». 3mm WW 0? 3mm Wmmma \$WW~ -7: WEN); 3.90:: 1*e+a‘+.;.+51?‘ 4.3151,“! . L2H f—Z’Lgxul J‘sil‘ﬂé’uhﬁéfi' , , \O/ , 33, 358mm, 81*- (1) ‘ 3mm 113:8 : 7, W _ . Wm, man‘mmca \$121M .1 3 his .& SEW;ka 02mm £2343 m. _ \$0. ml'9~+a‘+~-*§W't "M é I" 7%“ " " 7 K-H 7 A 3V: VKLM lnbaf‘uﬁgk = a "/34 - \avxa K‘Wo“ - 0— no ' h Mn, L1) ELQ*&5,-.é£=1ln ‘= ‘9 “[34 gm gmmamm 1 - .— n “\$931+. .'f§m lea?) ‘—' _ +& a” ’ > ‘ 2 any: + énam—O =ana—HaWT 31"“ : amm. a} L,‘ :94 3' ” ‘ a~ l l C T x _ . mg g0 mm no such mm; , \$=ID em “ma ..+c3"’1+a“"+d“,= r . meme» Lt WNW} = WWW “3 «N? 9.63 M man 0? swam ﬂaming? W Mex smmak mums LU \$‘E’XGN1‘13ELTME’S wwwmx 84¢ m MS- meélmiqn ,. ‘5 ma; :1 gmxw WNW 3w m . 30 03 2452016“ 3 qu’VVL‘VLO ) \$01603 wax Vﬂb-M Mn, (1) {no-0.3 40°33 Y?‘3n£-3nfi c hf «Zyg fgmﬁ + l ¢ “3 f no; +3“: *3vw ‘-‘ (“9‘03 . 1W8 mam (1) ﬁb‘ﬁ‘aﬁ gen. gamma“ mwm’e Y\o 7M5 s==® QM n} < 0w)3 Vman. .5.propo§rrm-. wax, swan is dmmm ML; . WWW WMﬁsmxm Wmmr / Wﬁmex wmw 5”"H :9 (91 form MN M 8-31 9 mm! I371; H.973 QM ewmxm \$4M) Wm. «53¢ W‘YM menaran ’Vvagwm <5 was 3 mm mm 5&3, no. 30 u) ﬁfwltil .15 m1 .IEW‘V‘K‘M , K4 no “a = (as! ..TaR3/K:no—I \ 2 ‘1’." 5 “a i— I : 23°“ 4) +' Wo/\ =5“); 6&5 = (wk-h25151) - Luxaﬂta'l) A (9X 1 t9,(k+H)L5"‘) Tm mm m, 33mm can m msxm mm W) S=¢ am.» 52"“ H '\s Q\m\$\m m u WX ., C .ummmm: m .2,+21+....+2“=2”*'~2_ max. 4 1203? m LY: 3mm WW ~-‘ )/ WTMQX swam 24.2% +Z.“"# Z‘W|~ . Met §=Emw1 l .2+_2?-+ +1.“ 1 2'"*'-z§ * Wampum am .. 51m wwS: ' wﬂm mmrmmqmwm .sumga swame Wk; m). 30 ® 2+z¢+ .+ 2%" +2Vh+ 2“°*'-2 . ?)\x¥v_\,<m 2+th ....Zv..._ W4 3%; wag-«1 , WC?) 2+ZZ+ ..* 2"“: 2m ‘2 Add 73° /O 7321+. .*2“°"* 2"° = 2"“ 2 *7.“ , \ 1. 2.2300 ’2- / = 2”°“ :2 ms ,3 Wﬁm’r Wm ms v0 mnemm¥ Vlo t Tm 8:4) and 73214 J2“ = Z“"—?,_ i% W VLVL'E)(1 Immm: m we in) ,Mm’rmﬁ aim s.,,813..._ske22o Burn “‘3‘ SHSf , i5.<%1,1,,..._,§<§dam m 25”?“ . 28‘“ m pm) 136W, 31%me :Vneim , 11‘ 8.,32, . ‘mey’eéo mnm+ _ sue a; , L,j%§l_.2_, mm, 3%. vv- 23‘ + 23*». 2‘" . gyms. 1w): Z°=l JTM . Mum MWQ :wae W*W£N mm \a: ﬁWnim 35 am. 0L , - mama: W3. oil- . , \mwmm pm)? Um) ﬁma m £6 6” ' mm 0 * _. - u; (WOEE Y‘;‘_/_z_ = \$181+ \$733K .. ﬂmeVxxM mm. n12" , 4w =2:c_2‘5'*291» 323“) = 235‘“ FEW. L iez‘i‘k" . -\} INQ£O W ME, .amvsimmm A x3 «Maw a??? mmckwm , , mm’rwg L 3:.) xv 23‘9sz ’nggmv 13‘ *ﬁ“ 42%“ H=18Mfw .~*2&“«?4°. . NB. \S. WSW, ‘0? 1'3 Bear (‘5: (3mm mm it”; 2 5M) 9L“;er gm )3. 8g maxan 051 Hammxwx Noam ,, mm is m 2M2 — ...
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graded hw 6 - Due Monday 31 October 2011 Homework 6\jff(l...

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