CMSC 250 – Jerry Alan Fails Student: __________________ Due: Tuesday, July 3, 2007 ID #: __________________ HW #8 Due: Tuesday, July 3, 2007 Page 1 of 2 HW #8 You must work alone on your homework, and homework must be written legibly, single-sided on your own lined paper, or typed, with the answers clearly labeled and in the sequential order as assigned. You must write your name and university ID number in the upper right-hand corner of your homework. Staple all pages together and be sure that your name appears on every sheet. 1. (-10 points if wrong) Write your name clearly on each page. Write the time and place of Exam 2. 2. (10 points) Suppose P(n) is a property such that i) P(0) ii) For all integers k ≥ 0, if P(k) is true, then P(3k) is true. Must it follow that P(n) is true for all integers n ≥ 0? If yes, explain why; if no, give a counterexample. 3. (70 points) For each of the following say whether it is true or false. If true, use mathematical induction to prove it. If false, give a specific counterexample. (a) ()138,21−Ζ∈∀≥ nn . (b) ++−=++−+Ζ∈∀∑=+2121211,1nniiiinni. (c) ()134,2−Ζ∈∀≥ nn . (d) −=−Ζ∈∀∑=+1231,2nkknnk. (e) 522,26+≥Ζ∈∀≥nnn. (f) nmnnm2,,3≤Ζ∈∀Ζ∈∃≥. 4. (20 points) Suppose a4, a5, a6, a7, … is a sequence defined as follows: ≥++====−−−4 integers allfor 2,1,0321321kaaaaaaakkkk (a) (5 points) What are a4, a5, a6, a7? (b) (5 points) What can you say about the parity (i.e. evenness and oddness) of ak, in general? (c) (10 points) Prove it.CMSC 250 – Jerry Alan Fails Student: __________________ Due: Tuesday, July 3, 2007 ID #: __________________ HW #8 Due: Tuesday, July 3, 2007 Page 2 of 2 HW #8 5. (No points will be awarded for this assignment unless this is done) Sign your name to the following honor code statement: “I pledge on my honor that I have not given or received any unauthorized assistance on this
View Full Document