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# graded hw 5 - Due Thursday Homework 5(1 For 7 E R{1 and n E...

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Unformatted text preview: Due Thursday Oct 20, 2011 Homework 5 (1) For 7' E R/ {1} and n E N use induction to show that: 1 —- n - 1+r+r2+---+rn"1= r. 1 — 7' (2) Prove by induction that for each n 6 N22, 712 < 713. _. IXM‘ f ,X < lxhﬂlx (3) Prove by induction that for each n E N, n3 + 2n is divisible by 3. ’— (4) Prove by induction that for each n E N, and for each an E 1R>2, ‘ VP?» 6 ’Xm‘+3<< “‘4 ﬂ. NW“ ‘*’ ~91 X L n“‘+‘>< ’34 xn+\$ < mn-\$.X x [X “I, < Mil ‘ \n ~ m ’ “X L' ~i_/X‘*/\/xw.,\kq‘ -\\ I] h (5) Prove by induction that for each n E N, and for each :c E R>2, m: + 1 < :13” + 2. ’5 (6) Prove that for each n 2 1, 52”—1 + 1 is divisible by 6. (7) Prove the rule of exponents, (ab)” = anb“, for every natural number n. (8) Let n E ®>0. Prove that n2 — 1 is divisible by 4. ‘ I x l” 1060 Wm! /zK+\)ﬂ*\:lw+qu\ : quLZmew owner 4 NH ' l ’ SZHJT 5 *lmWX/Wﬂu. Woo“! xi Lg Psm‘kani‘JB-BM-xbljﬁth \l __ 5”“."”‘ m 25( Loki") 4 a a 25LM)[email protected]) V4147”, L“ M}; u“ . .7 I, \ J Lam-52%;? Y’e/‘R/ix}; 3m max 9&\mgmrx©sm may: “manuayw‘. , F" \"( . M, pmqu mmwca \Mwnm: n , W3" WW“ ..._+Y““ = H/H _ ‘» 93mm pm: MASH Ray “Ar =1 (we ' \9 we)! mm: m0 ﬁwﬁmmxx mm mm mm \mmm Mp’rmsg 2. m9: 4900 Mm ﬁx me n21 ; mm“ W 1 30910 320A) = an+\\_ We WyD is m. {awxismj LI“ ' ’“ PM“): "*“Y’h 3 "3' (WWW : ‘4 A” \ ‘m ‘1; Bummmwmpmm \+Y+Y‘¥..,*YV‘“ = H ,*&QQ\Y“TO€§1<3\§LM + k ‘. ﬂ“ : \-Yv‘/_ 1- A t mf‘ Y ‘qﬂignmg - 14mm“ *1 2 “WM . ,“mmx‘ Mkr ' FY . ﬂ \‘Y + ma k \*v+v"+x.,.*r““= "" M 30,9an émexxAJmammvm mm 32.9me *1 nix, # ,2 ..p{ap5wm-. mm m \mmmmm ma Y\ a; N22. Yxl< v6” / 32m; mmmm mm: mm r., WW9 . 7 \ 8&sz pm; 7f<291¢> LMa “WW 7 {o mm“ mm\; «1 ABM malv gm), \3. W W n21 ‘ :/ /’ \MMW 5m. m pm.) 32mm) «mm gm) :3 Wﬂ‘édmmmnﬁﬂ‘ ” (m\)‘<(h+\7’ ‘3‘ n1+2n+\ < n3+3n1 + 3n+_\ - / w «m mm mmsxs w “’5 ~ 9m 7m “VD. mang \ . man +1< n9 r mm m m: mu m mm mm: «exam s jg m » WWHNQWEWL, _ ,, A . go‘ g») épwﬂﬂm m. mm mm; m: ma; . - / JUAN“ £5. , "movvpmm 23m “EN 4 “Quin XS dwxswyoﬁﬁﬁ , M M \$019), t Main: 3%. 7632KQ88£=1XQ t\13+2m_= 3: BBQ x/TM _ mom mmsxs: Wm»; mm mm £0; 53m m,t,,/ _ {r 1 mom t makava 11mg xsmmxwrwx B‘réﬁﬁmm- + (mY’Jrle) = n5 *2>v\“*5n*2>, , 7 B m, \mmme m n?* m +— 3” * 3m +3. w W ./ mgr/W 3Y\_ 30Wpr A _ MWW‘HFREL WY! MM = MHNG? ﬂaiKukKHo'? Mon/m :s’DLmﬂ. /' mommaposam \Mmmmyzacn mexmfomacn kmemn'x 1mm m MMMsm‘t mcmm - mm}: «W ‘ QK“"X 8831 tag; 1 1pm “MUX-X 1M3? 2% m2: Mxe’ﬂm : \‘6 *I’YWN ’903 SW em QL‘M émnﬂ), ,mvx mm \37 ’éﬂéﬂ. ; Manama WW5: Asswmmgk thxm W “3t ' L Jnduttxxtgﬁﬁv Swank Dorm) ,wws: mm) is m 1, 7 J; pm“): xnn T ,X < ,XQMJX /Xn+l < Kn“ 9 + '34 mm mum mpomxs pm}: MW XW + “4 < «W-‘f w W Mac,th “Vi Xmﬁ‘ «Mt/’9 1”“ mvwmm “Wm “45" mm safgcwm V8\\dl‘§‘dfﬂk\wp\®~j ’L W“ 9mm mmﬂfif 1K ’1< «TMva oflcimmwndmm ’ 1““ 9Vm< Ix“*1EP(n+\) L 33.W)=5pm+1)=mus, WWPMI yon) MM kiwi vb/ 5~ ywplsmm: ma m maxmm ww 26cm ner. an mach «ﬂu mw < J“ +2 ’. mm? m Wmmw \Mmm: 12* gm Ska’xﬁvmm WM 11"“ VneM}.’L‘Kn Emcaxzmx): X+\<~’,X+Z , t<2 \me, WW \51 \MeN 'pLx)\%M am V043 évmn) ‘mn pm} 13 N \mmz NW8“; ASSN max W0 km: ﬁx 83m m - »_ m WW‘ mum SR9; mm mm whim) Wm. pmﬂ kw 3’ ~ WHY mm H k 1"“ +1 mm mam “mm mm: ma < 1.“ +1 add. 'x w mm M: + t‘. “MM 4 mi‘vxw ﬂvxwumwmvwxwmd m“+x<0\$“-¥ :1 NM (HWQH 4 %“~'I\+L (“'W‘ “Me, mg; [73% «0 \$4 mmmu .33" .3 + ‘> (mum < 3"“ * 2 Wm «\mmquaimmum. _ A i so. pm»:an Tm mmw v mm MM Wm ;b_-pqwtmgvmwmx 4b: Q60“ nix _, 51"" +\_‘\§ (imam m (:2 ; mm: Assm 3» mm \_\o\45”" *1 i 5“ = w ’\ ,45 e 9;+ A} xv m ; 5.1“1 +|= 5W'1+\_= zBMvwh zsuslﬁ‘m = K9, (1543-14) ._ \2\ “1515143: 2513'“ Q N: Tm; SEAN Mills 1. (mm (z 1. m1 WM Lab)“: 3%“, for WM nawmyxmwyx mm immmi L9: pm) mm mwm (anT=a“b“ . Bax idﬁr‘: \$.03:- awab {TM \C. We K pm is,“ gm 9m). \$120.4“) gm 9an ism W 2N mdwn'vi mm»: Asgwm mm iaw Wm WM . b \MMM Wifsmo pm) % pm”) ,mm PLWM) 15mm».me mm; 1mm: Lab?“ = a“ DVM MW \mmz nqmm may. Mb)“ = a“ b“ * N‘ch Wm b4. ab Lab‘iéb)“ = a“ b“ - ab (aby‘m ._ (ﬁn-anﬁ ‘ *Mg 5%» \5 {waxes/xv :9 31mm) Thus, m m. WNW bi kaaL mm pm: mwa yum km w n X- \mpxmmt m “0.59.. n14 B d\_\m\‘0\1 114 l} 1 ; a gm) (Wmth an QM mmW ‘Y\ = ZVLTH ,_ VLQ'M r "3 v94, = mm)? —x = w Mcth / ‘ “H = HWMX e N . ms 4&1vath Wm, M \ Mm‘whs‘dmf AWN n§Q>O . T s) ...
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