hw08 sol

# hw08 sol - Spring-2011 CMSC 250 Homework 8 Due Wed Apr 6...

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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 n/ 2 + 3]. Show that, for all n 1, a n 1 (mod 4). SOLUTION TO QUESTION 2 BASE CASE: a 1 = 1 1 (mod 4). a 2 = 21 1 (mod 4). INDUCTION HYPOTHESIS: For all 1 m < n a m 1 (mod 4). In particular a n - 1 1 (mod 4) and a n/ 2 1 (mod 4). INDUCTION STEP: a n = a 2 n - 1 + a 3 n/ 2 + 3 1 2 + 1 3 + 3 5 1. 3. (25 points) Define a n as follows: a 1 = 1, a 2 = 10, a 3 = 500, a 4 = 1000 and ( n 5)[ 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.

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