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Unformatted text preview: CMPSC 360 Discrete Mathematics for Computer Science, Fall 2008 Homework 2 Out: Sept. 4, Due: Sept. 11 by 5:00pm Instructions: Write your name, section number and “CMPSC 360” on your assignments. Put your solutions in the homework box in 342G by 5pm on Thursday. 1. [Practice with induction] Prove each of the following statements using induction. (a) For all natural numbers n > 1, 1 + 1 4 + 1 9 + ... + 1 n 2 < 2 1 n . (b) For all natural numbers n ≥ 1 , 4 n +1 + 5 2 n 1 is divisible by 21. (c) For all natural numbers n , n X i =0 i 2 = n 2 4 , if n is even; n 2 1 4 , if n is odd. [Here for any real number x , b x c denotes the largest integer less than or equal to x .] 2. [Recursion and proofs] Consider the following computer program: function G ( n ) if n = 0 then return 0 if n = 1 then return 1 else return 5 G ( n 1) 6 G ( n 2) Prove (using strong induction) that for all inputs n ∈ N , the value returned by the program is G ( n ) = 3 n 2 n . 3. [Research project?] One day, you find a shadylooking flyer on downtown that advertises an undergraduate research position in the Ohio State Geology department. After you make your way down to Columbus, Dr. Hoover, the professor in charge, hands you the following asdown to Columbus, Dr....
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 Fall '08
 HAULLGREN
 Natural Numbers, Mathematical Induction, Natural number, Dr. Hoover

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