Stitz-Zeager_College_Algebra_e-book

One of the goals of this section is describe the

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 = a (b) j =1 1 − rn , if r = 1, 1−r n arn−1 = na, if r = 1. j =1 3. Prove that the determinant of a lower triangular matrix is the product of the entries on the main diagonal. (See Exercise 5 in Section 8.3.) Use this result to then show det (In ) = 1 where In is the n × n identity matrix. 4. Discuss the classic ‘paradox’ All Horses are the Same Color problem with your classmates. 9.3 Mathematical Induction 9.3.2 579 Selected Answers n j2 = 1. (a) Let P (n) be the sentence j =1 n(n + 1)(2n + 1) . For the base case, n = 1, we get 6 1 ? j2 = j =1 (1)(1 + 1)(2(1) + 1) 6 12 = 1 We now assume P (k ) is true and use it to show P (k + 1) is true. We have k+1 ? (k + 1)((k + 1) + 1)(2(k + 1) + 1) 6 j 2 + (k + 1)2 = ? (k + 1)(k + 2)(2k + 3) 6 k (k + 1)(2k + 1) ? +(k + 1)2 = 6 (k + 1)(k + 2)(2k + 3) 6 j2 = j =1 k j =1 Using P (k) k (k + 1)(2k + 1) 6(k + 1)2 + 6 6 k (k + 1)(2k + 1) + 6(k + 1)2 6 (k + 1)(k (2k + 1) + 6(k + 1)) 6 (k + 1) 2k 2 + 7k + 6 6 (k + 1)(k + 2)(2k + 3) 6 n j2 = By induction, j =1 ? = ? = ? = ? =...
View Full Document

This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

Ask a homework question - tutors are online