hw5(2) - a n < 2 n for all n > 0. 6....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
CMPSC 360 Discrete Mathematics for Computer Science Fall 2009 Penn State University Assignment #5 Due: 28 September 1. Let n be a positive number. (a) Give an expression (using summation notation) for the sum of the first n odd integers. (Note: this is not the sum of the odd integers n .) (b) Prove by mathematical induction that this sum is equal to n 2 for all n 0. 2. Consider the following sequence: 1 , 6 , 11 , 16 , 21 , 26 ,... (a) Give an expression (using summation notation) for the sum of the first n terms in this sequence. (b) Prove by mathematical induction that this sum is equal to n (5 n - 3) 2 . 3. Prove by mathematical induction that for every integer n 4 there exist integers p and q such that n = 2 p + 5 q . 4. Prove that n ! > n 3 for all n > 5. 5. Consider the following sequence: a 1 = 1; a 2 = 2; a 3 = 3; And for k 4, a k = a k - 1 + a k - 2 + a k - 3 . Prove using strong induction that
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a n < 2 n for all n > 0. 6. Prove using strong induction that any propositional formula is equivalent to one that does not contain any uses of , , or . (Hint: recall Problem 9 from Assignment #1.) 7. Prove using strong induction that for any propositional formula P that is in negation-normal form, P is equivalent to a formula in negation-normal form. 8. Consider a family tree that consists of just you, your parents, their parents, etc. (no siblings). Assume you are Generation 0 , your parents are Generation 1 , your grandparents are Gener-ation 2 , etc. How many ancestors do you have at Generation n , for any n 0? Give a proof using induction....
View Full Document

This note was uploaded on 10/04/2009 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Pennsylvania State University, University Park.

Ask a homework question - tutors are online