mid2_07_sol

# mid2_07_sol - 1 11 JIII W V lTota11 l i MATH 204 Midterm 11...

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Unformatted text preview: 1 11 JIII W V lTota11 l i , MATH 204 Midterm 11 Duration: 90 minutes Show your work 08.05.2007 Name, Surname/Student No: Recitation Section: 1. (4pnts) Prove that 2:; (33* = 4". (Hint: 3+1 = 4.) ‘9: (BMW: icmzL-WL as 4m (embed Than“ IA-=O O z z. (3):“. \A—ro 2. (5 puts) Find an explicit formula for the sequence (a,)n20,which satisﬁes the relation an = 50H —4a,._2 +3 '2",Withau = -6, a] = 0. &A: {ha-i— '1 kW -1 ' (anst hm. ko-1guwv\ VLKAbM 2 an», PAOA‘L 5m; xg 3M4 —, <x-l\Lk__L1) ﬂ r1: 1‘ wt“ 1 ‘ SQ": “I an 0A fakiélﬂf V (9"?‘(W 4.. ?u\\'~¢va 5-L..l'\'\~ r33 ’H/K P’suwvb (9) Fc-LR‘VM\ 4“ ’— 4'9.“ 5) 4:2“: 5"; 2A,: (kc-2A.; 3-2.“ (1L5 L9C~z1¢4k ll 20 Cc-‘é. Runner “Wk hk~kxw~ l5 », 61A: «59.6% = «(111-4 —é—Z wfii'k JHNL \M5H\ Univ”; 05.: “(9: a“+JL_é 1:) (XL-(41:0 I dlz-dL q‘;°!\+('“lz"n' to a) 0(qu 1 11:’L‘l- ;) q“: WH+ (1A*4~ sr‘i n 0&4: "1+ ‘1” - 3' 2“” 3. (4+2 puts) a) Suppose that we select 11 (distinct) integers in the set {1,2,. ..,99,100}. Prove that there are at least two integers x, y among those we have selected, which satify lfi — 07 | < 1. (Hint: Consider [It] for suitable k.) LO; gnaw «(A knee“, ‘1’ ___)6M_ Sm-u ‘5‘“"W‘mil‘w, (éJZM,-, ﬁMeo I»; Isl—4&3,” , L4“: 5 UL km“ we. km. A« “Scat-5 C max, urns) 1.. \p‘, 9e“; au— lo 5.“: C 4», #719) ,wbvcnk, éwku 50 HM Pkgakh‘h Y'Nu‘eu. +ka I at» (wt- l 6 «HM was)", LEA) \$40 LJTJ [LG] m w» h,» saw—e. “lax L M «m worés : , Thts Mm; (sepa— K=t-U) lf:_f6 \,< ,1, b) Write a statement that generalizes the result of part (a). (No partial credits!) L; E A2 ’2 19 g a A a, £64, Su '3‘: as... Aw“. saggut. Ml Acskaal’r Nag“ M 1:: \$43: 3 t) “,1 A15, =7 A“? \m'\ 4M“; “Ad—2:4“: Van-was ‘hm— S‘l‘tmkl Docs saH‘s‘go ; lﬁ’ral 4L 4. ( 3+5 puts) Recall that a bit string of length n is a sequence (c1 ,02, . . . 0,.) where c.- 6 {0,1} fori = l,2,...,n. Suppose that 0,. denotes the number of bit strings of length n, which contain two consecutive zeros, where n z 2. 3) Find a; a3 and a4. SAC/Sum «a 2.— , 2— WﬂhgﬂukUl— Neg \‘ (ale) ‘7 “l a 3/»r‘ : (0'9)-\\ (pic/g\((fs/g) l4 4 a A) « ﬂ 2‘. alo’o/cﬂ L old/4’ °\ ( viola, L\(oloj «’4‘ ( LO" 9 D\ I / / Cl;o'¢'°\l ({lﬂlo/l‘)((/A)OIO)4 b) Find the recurrence relation, which the sequence (0,.) satisﬁes for n 2 2.(Hint: A bit string of length I: either ends with 0, or with 1.) A gcﬁhuu» /B kmg‘ji'“ VHLUQOC‘A A21) a") om. +\NL golkewvm lags a was’ £4!» 1 ( W, \o..§¥ (l A ‘HN. 195'? +V° 4—3 4“ ~& \5 H 2 '4 ‘1) anés lo)" ‘\ 9 DC) (A mrw 7. can-W's Thu—rt are. 41W” 5aju-ru~ 78 +afc L1 14 Al (L'AA “ ‘ a ‘1 has 2—- cansﬁue'i‘JC-v Eugenia.» ckaee CL.Q.\ Equa 2A M. was. Thug-t an wqula {a " —t Z—A A21, ana'z,’ «A—tAtGA I 5. (3+4 puts) For all n 2 1, we define n = n =n,andfor2£k£n, n+1 = n + n . 1 n k k k—l 6 a) Determine < k > for 1 S k S 6 explicitcly. (Hint: Use an analogue of the "Pascal’s triangle"). < i 5 Mm, +rig¢3\¢’ 5-lL-~u <25) %,> gr” Arm, Mm‘e‘“ <3.» <9 42;; (D: * <Cz> <26 #37 <95 <2-» <:.> \/ l . < “:37 ‘ L » <a>=<§>se 29. '3 ‘4 3 L, ‘} Q L! :54 3",“ H1 LL g- ‘ <‘3\}:<’é’\$: 614 1r zr-re g__,,) , a, b) Find a, b satisfying: n = a n + 1 + b n — 1 , where n denotes, as usual, the binomial k k k — 1 k coeﬁicients. Prove your result. (No partial credits!) "(ML (“\Qkop Show“; gar A A (<L< / M ~ ~ '\ / M var—K cu (tr { 1s <EB=ar<%>+b<;\$=\$q«h w 2-» <a>= « <1>+b<a>~m+¢~ 5:.( We; u.) e. Lu... 5% «\P \) row. Mk‘ 9 (A:\>v(n~ij 3a”. a“ A21! (gkéw‘ la. »\ Us»: Twéugb‘na g.“ A . Vii—7." 1 9.: < 2.- “ (1 A 1‘ z w i . ' . - a a” —<:Qﬁ>— (g "(1 V (begun) k) 3 3 A sow-k 69 mi; g» M A ,Mw-xu 25m”) n-t‘ »_ ‘4 ‘ k i >—i <' a * 91> : “Sb-(21%(1’153~("") Mb“ k.» Wu Les; i ‘ M‘JP.+&LsB ("SSH CTSB— ( l \1 (:13; / 5.? 1’ y; “MN” ‘2 S i4, ,5 A “(13% ( A ‘t L C a: \cLAA,X'{ ‘ \M ' ' \ 4) Thu mta k3“) lg:n‘f¥ MV\+ Le. C.®“3K‘«ld’ﬂ A ‘ 33;: N, .5 ar we L; m sq). sign A , Sivan. ‘m‘gux’1rm E v a«« x a«2 a- a w 1 («W L a “*R a i‘ A if L \ Wmmmwﬂ~ “‘7' ‘ w‘ ‘- \ ... . , ﬂ ( 5. R g _ ‘\ pf" )' KN n4": '/Iiﬁ‘(i ( A-t’k\\£ ‘- 3A ‘2' \$51 ...
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mid2_07_sol - 1 11 JIII W V lTota11 l i MATH 204 Midterm 11...

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