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Discrete Mathematics with Graph Theory (3rd Edition) 133

# Discrete Mathematics with Graph Theory (3rd Edition) 133 -...

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Section 5.2 131 10. The first few tenns are 1, 2, 3·2 = 6,4(3·2) = 24, .... Our guess is that an = nL When n = 1, 1! = 1, in agreement with al. Now assume the fonnula is correct for n = k; that is, assume that ak = kL We must prove the fonnula is correct for n = k + 1; that is, we must prove that ak+l = (k + l)L But ak+l = (k + l)ak = (k + l)k! = (k + 1)!, as desired. By the Principle of Mathematical Induction, the formula is true for all n ~ 1. 11. [BB] The first few tenns are 0, 1,0,4,0,16, .... Our guess is that { 0 ifn is odd an = 4 ¥--1 if n is even. We will prove this using the strong fonn of mathematical induction. Note first that al = 0 and d- 1 = 4 0 = 1 = a2 so that our guess is correct for al and a2. Now let k > 2 and assume the result is true for all n, 1 ~ n < k. We must show that the result is true if n = k. If k is odd, then ak = 4ak-2 = 4(0) = 0 since ak-2 = 0 by the induction hypothesis, k - 2 being odd. If k is even, then k-2 1 k-2 k 1 ak = 4ak-2 = 4(4- 2 - - ) (using the induction hypothesis) = 4-2- = 4'2- as desired. By the Principle of Mathematical Induction, the result is true for all n ~ 1.
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