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M122F07MT3Key

# M122F07MT3Key - Math 122 F01 2007 Midterm 3 v 3 Name QM The...

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Unformatted text preview: Math 122 F01 2007 Midterm 3 v 3 Name: QM) ' The only Calculator allowed is the Sharp EL—510 R. This question paper has five pages and 10 questions. For each question except 1 and 5, Show enough reasoning that it is possible to determine Where your answer came from. Don’t write that it came from your neighbour’s paper! A total of 43 marks are available. 1. [3] Give a recursive deﬁnition of the set X of all non—negative integers that can be written as a sum of fours and ﬁves. 0e X ' Iii? 'RQVQ iiida‘wv ><+>+e74 0m} Misc-:X 2. [3] Find the positive integer a if gcd(a, 630) = 105 and lcm(a, 630) = 242, 550. (030 on :1 \O 6x1q256© 3. [3] Find the base 3 representation of 512. 110,. '2‘ L5 is 9. j i r” ’2, 3 LE; 0 3 L2; 0 O 60. 1 (Zoo Ail/NS /1 3 3 .3 4. [3] Is the graph G' = ({1,2,3,4,5,6}, {{1, 3}, {3,5},{5, 1}, {2,6}, {4,6}, {3,6}}) connected? i 5. [4] Let f : 2+ ~+ R. Circle all true statements. No justiﬁcation is needed. @x + 2 + . . . + n is 9(712). If lirnnﬁm log(n) exists and is ﬁnite then f(71) is O(log(n)). (c) 71.! ~ n” is 0(2”). @0713 ~ Qﬁ + 7l0g(71)‘ 21 is O(n"). (6) If f(n) is not @(ng) then ﬂu.) is not 9012). (f) If f(n) is C(n”) then ﬁn) is not 0(71 l0g(n)). @f f(n) is the number in {0, 1,2, 3, 4} to which n is congruent (mod 5), then f(n) is 0(1). 6. [4] Use arithmetic (mod to show that 3 | (7" — 4“) for all integers n 2 1. viii (maxi Cwmci 44:3 l imcifl 3\ A i/\ W (A i l __- ‘ t ‘7 <4 i \\i. 20 (meal %\ wiMO (.v. a .V\ V\ I "slim sin/o (WW 7. Let the sequence a0,a‘1,a2, . .. be deﬁned by a0 : 1, a1 = 3, and an“ : Lian — 4an_1+1 for n 2 1. (a) [2] Compute 04. (b) [6] Prove by induction that an 2 'n. ~ 2” 7|— 1 for all n 2 O. N \ .\ _ r, “gmﬁ A“; (\:03 Q©:\:U~2,~\\x/ IQ v“\:‘\1 CM :3 1 Tina \$ﬂi€M€n¥ "r: Away. w¥wn n20 OW‘ \u2\+\.\/ (\33 \$3: Assam ngovfw ) QR vim PW; xiv 80M kfzm V 3%, l“;. A 3; I 4 m aka/p3? GK“ '4qu 4470\wa _\(\ . _ K . Kx‘ ‘1 “V UV CL MW ~ L¥(U/.’~\\)‘L EA, +\ _ 7‘ ‘z‘\ “ LR KZRFArK‘K—D'Z: «+\ :1 1+2“ Law» LwﬂXH 2 1:4 “1 Z * Z MGM 4: \ /3 8. [4] Let (1,!) and c be integers. Prove that if a l b and b l c then a | c. gKan {(-‘l/ r“ 3 OK\A 9‘ [4] Use the deﬁnition to Show that n" is not 0(722). . ﬂ L EA O m \ L: a; R3233: Gig/{L4}?! VHS" C: > f) .4“ ﬂ 1 . ‘ ‘ﬂ £d“‘(\ \ . ‘ I JFLM‘OL“ V\ mc‘i’i 23/ N5 CAN 3 ) m2» "\ 1 ﬂ {1 ﬂ ': 5: Qn :"C/ fir-7&3: ml. [AZ 10. [7] Solve the recurrence relation a0 = 0, a1 = 6, and an+1 2 6%4’ — QaEHz/for n 2 1. i/\ ﬂ" R €30?ch m A ‘ w A QAQAD. 0mm ’ (OCH/WV qq’mq ’“ U I/EN‘, ...
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M122F07MT3Key - Math 122 F01 2007 Midterm 3 v 3 Name QM The...

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