Stitz-Zeager_College_Algebra_e-book

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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 ? = ? = ? = ? =...
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