Assignment_5 - n> 4n for all integers n ≥ 0 6 Let b 1...

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

View Full Document Right Arrow Icon
Sequences & Induction Due: September 29 1. Let n be a positive number. Give an expression using summation notation for the sum of the cubes of the first n even integers. Then show that this sum is equal to 2n 2 (n+1) 2 for all positive n. 2. Consider the following sequence: 2, 12, 22, 32,... Give an expression using summation notation for the sum of the first n terms in this sequence. Then use induction to prove that this sum is equal to 5n 2 - 3n for all positive numbers n. 3. Consider the following sequence: 0,3,8,15,24,35,48,... Give an explicit formula for computing the k th number in this sequence. (Hint: it only depends on k.) Then use induction to prove that the sum of the first n numbers in this sequence is equal to (2n 3 + 3n 2 + n - 6)/6 for all positive numbers n. 4. Prove by mathematical induction that 2 n < (n+2)! for all n 0. 5. Let a 0 =6 and a k =a k-1 +4 for all k>0. Prove by induction that a
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: n > 4n for all integers n ≥ 0. 6. Let b 1 =3, b 2 =5 and b k =3b k-1- 2b k-2 for all k>2. Prove by induction that b n = 2 n +1 for all integers n ≥ 1. 7. Suppose we have N deck chairs on board the Titanic and it ʼ s our job to stack them all together into one pile. We can take any pile (of one or more chairs) and place it on top of another pile to create a new pile. Assuming none of the chairs is initially stacked on any other, how many steps (a step is the act of creating one new pile out of two) does it take us to stack all N chairs? Give a proof using strong induction. Then go hop in a lifeboat. 8. Prove using induction that the following statement holds for all n ≥ 0: if n is odd then 9 n mod 10 = 9 and if n is even then 9 n mod 10 = 1. COMP SCI 360 Assignment #5 Penn State University Fall 2010 PAGE 1 OF 1...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern