ex2 - CS70 Discrete Mathematics for Computer Science Spring...

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CS70 Discrete Mathematics for Computer Science, Spring 2012 Homework 2 Out: 26 Jan. Due: 5pm , 2 Feb. Instructions: Start each problem on a new sheet. Write your name, section number and “CS70” on every sheet. If you use more than one sheet for a problem, staple them together (but do not staple different problems together). Put your solutions in the boxes on Soda level 2 by 5pm on Thursday: your solution to Q1 goes in box CS70–1, your solution to Q2 goes in box CS70–2, and so on ( five boxes total). You are encouraged to form small groups (two to four people) to work through the homework, but you must write up all your solutions on your own. 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 , 3 2 n +1 + 5 n - 1 is divisible by 4. (c) For all natural numbers n , n 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 , x denotes the largest integer less than or equal to x .] 2. [A tiling problem] You are given three kinds of tiles, A , B , and C , of dimensions 1 × 2 , 2 × 1 , and 2 × 2 respectively (as shown in the figure below). Note that rotations of the tiles are not allowed, so tiles A and B are not the same!
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