{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw5(2) - a n< 2 n for all n> 0 6 Prove using strong...

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

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
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

{[ snackBarMessage ]}

Ask a homework question - tutors are online