Discrete Mathematics with Graph Theory (3rd Edition) 113

Discrete Mathematics with Graph Theory (3rd Edition) 113 -...

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Section 5.1 111 as desired. By the Principle of Mathematical Induction, the result is true for all n ;::: 1. (c) When n = 1, 12 + 3 2 + 52 + ... + (2n - 1)2 = 12 = 1, by definition, while n(2n-1~(2n+1) = 1(2-1~(2+1) = 1 and so the result holds. Now suppose that k ;::: 1 and the result is true for n = k; that is, suppose 12 32 52 (2k 1)2 _ k(2k -1)(2k + 1) + + + ... + - - 3 . We wish to prove that the result is true for n = k + 1, that is, that Now 12 + 3 2 + 52 + ... + (2(k + 1) - 1)2 = [12 + 3 2 + 52 + ... + (2k - 1)2 J + [2(k + 1) - 1J2 k(2k - 1)(2k + 1) ... = 3 + (2k + 1)2 (by the mductlOn hypothesIs) k(2k -1)(2k + 1) + 3(2k + 1)2 (2k + 1)[k(2k - 1) + 3(2k + l)J 3 3 (2k + 1)(2k2 - k + 6k + 3) (2k + 1)(2k2 + 5k + 3) 3 3 (2k + 1)(2k + 3)(k + 1) (k + 1)(2k + 1)(2k + 3) -'----'-'---'-'---'-=-'---'-'-----'-"'-----'- 3 3 as desired. By the Principle of Mathematical Induction, the given statement is true for
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Unformatted text preview: n ;::: 1. (d) When n = 1,1· 2·3+ 2·3·4+3·4·5+··· +n(n+ l)(n+ 2) = 1· 2·3 = 6, by definition, while 1(1+1)(1:2)(1+3) = 1(2)~)(4) = 6, so the result holds. Now suppose that k ;::: 1 and the result is true for n = k; that is, suppose that 1.2.3+ 2.3.4 + ... + k(k + l)(k + 2) = k(k + l)(k: 2)(k + 3). We must prove that the result is true for n = k + 1, that is, that 1·2·3 + 2·3·4 + ... + (k + l)[(k + 1) + l][(k + 1) + 2J Now the left hand side of this equation is (k + l)[(k + 1) + l][(k + 1) + 2][(k + 1) + 3J 4 1·2·3+ 2 . 3 ·4+ ... + (k + l)(k + 2)(k + 3) = [1.2·3 + 2·3·4 + ... + k(k + l)(k + 2) J + (k + l)(k + 2)(k + 3) = k(k + l)(k + 2)(k + 3) + (k + l)(k + 2)(k + 3) (using the induction hypothesis) 4 = (k + l)(k + 2)(k + 3) (~ + 1) = (k + l)(k + 2)1 k + 3)(k + 4) as required. By the Principle of Mathematical Induction, the given statement is true for all n ;::: 1....
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