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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
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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.
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“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 \
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ml. [AZ 10. [7] Solve the recurrence relation a0 = 0, a1 = 6, and an+1 2 6%4’ — QaEHz/for n 2 1.
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 Spring '11
 stephenlang
 Calculus, positive integer, QR vim PW, UV CL MW, non—negative integers, Sharp EL—510

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