Discrete Mathematics with Graph Theory (3rd Edition) 143

Discrete Mathematics with Graph Theory (3rd Edition) 143 -...

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Section 5.2 141 for all m, 1 ::::; m ::::; k. There are two possibilities: m = k and m < k. In the first case, gcd(Jk, fm) = gcd(Jm' fm) = fm = fgcd(m,m) = fgcd(k,m) which is the desired result. In the second case, we have by part (b), hence, gcd(Jk' fm) = gcd(Jm.' fk-mfm-I) by Lemma 4.2.5. By part (a), we know that gcd(Jm, fm-I) = 1. Thus, gcd(Jm' fk-mfm-l) = gcd(Jm, ik-m). (See Problem 11 in Chapter 4.) Since m < k and k - m < k, we may apply our induction hypothesis to whichever of these is the larger. Since gcd(m, k - m) = gcd(m, k), we obtain gcd(Jm, fk-m) = fgcd(m,k-m) = fgcd(m,k) as desired. 57. (a) For n = 1, Vn - Un = 1 - 0 = 1 = 1/4°, so the result is true. Now assume the result for n = k. Then Uk + 3Vk Vk+l - Uk+l = 4 Vk - Uk 4 1 4k which verifies the statement for n = k + 1. Thus, Vn - Un = 1/4 n - 1 for all n ~ 1. Un + Vn Un + Vn - 2u n Vn - Un 1 1 (b) Un+l - Un = 2 - Un = 2 = 2 = 2 4 n - 1 > 0, so Un+l > Un, for all n. Therefore, Un is an increasing sequence. ( ) _ Un + 3vn
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Unformatted text preview: _ Un + 3vn -4vn c Vn+ 1 -Vn -4 -Vn -4 decreasing sequence. Un -Vn 4 1 1 ---- < 0, so Vn is a 44 n -1 (d) Ul = 0 = ~ -i;rh, so the result is true for n = 1. Now assume it's true for n = k. Then Uk+Vk _ Uk ~( _1_) 2 -2 + 2 Uk + 4k- 1 1121111 Uk + 24k-1 ="3 -64k-2 + 24k-1 2 1 1 3 2 1 (4 -3) 2 1 1 "3 -6(4k- 2 -4k- 1) ="3 -6 4k- 1 ="3 -64k- 1 which is the result for n = k + 1. By the Principle of Mathematical Induction, Un = ~ -i 4}-2 for all n ~ 1. 58. If A = raj is a 1 x 1 matrix, det A = a. Now suppose A is an n x n matrix whose (i, j) entry is aij' Let Mij be the (n - 1) x (n - 1) matrix obtained from A by deleting row i and column j. Then n detA = 2)-I)i+ 1 ail detM il i=l (We have given the definition by "expansion of cofactors of the first column." Equally good definitions can be given for expansion of cofactors of any other row, or column, by making appropriate changes in the two occurrences of the subscript i1. See an appendix.)...
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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