# hw08 - a n = a n-1 + 2 a n-2 + 3 a n-3 + 4 a n-4 ]. By...

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Spring-2011 CMSC 250: Homework 8 Due: Wed Apr 6, 2011 NOTE: Due AT THE BEGINNING of Recitation! COURSE WEBSITE: http://www.cs.umd.edu/ gasarch/250/S11.html (NOTE- there is a Tilda before gasarch. It may be easier to get to the course website via the dept webpage (go to class webpages) or gasarch’s homepage (it will be obvious).) 1. (0 points) What is your name? Write it clearly. Staple your HW. What is your SECTION NUMBER. WHERE and WHEN are you taking the SECOND MIDTERM. (SEE shortsyll to see WHERE.) If you cannot make the midterm time and have NOT been in contact with Dr. Gasarch about it then contact him ASAP ([email protected]). 2. (25 points) Deﬁne a n as follows: a 1 = 1, a 2 = 21, and ( n 3)[ a n = a 2 n - 1 + a 3 d n/ 2 e + 3]. Show that, for all n 1, a n 1 (mod 4). 3. (25 points) Deﬁne a n as follows: a 1 = 1, a 2 = 10, a 3 = 500, a 4 = 1000 and ( n 5)[
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Unformatted text preview: a n = a n-1 + 2 a n-2 + 3 a n-3 + 4 a n-4 ]. By constructive Induction ﬁnd Natural numbers A,B such that ( ∀ n )[ a n ≤ A · B n ]. Make B as small as possible. Given that value of B , make A as small as possible. 4. (25 points). On the planet Gork the only currency they have is a 5-cent coin and a 6-cent coin. (a) Show that 19 cents cannot be formed using Gork Currency. (b) Show that, for all n ≥ 20, n cents can be formed using Gork Currency. 5. (25 points) Let S be the statement: If T is a full binary tree then the number of leaves is one more than the number of internal nodes. (a) Prove S by WEAK induction. (You MUST do the proof by REMOVING leaves from the tree, as done in class.) (b) Prove S by STRONG induction (as done in class). 1...
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## This note was uploaded on 12/04/2011 for the course CMSC 250 taught by Professor Staff during the Spring '08 term at Maryland.

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