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Discrete Mathematics with Graph Theory (3rd Edition) 151

Discrete Mathematics with Graph Theory (3rd Edition) 151 -...

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Section 5.4 (b) We have as desired. 2x[c(n -1)x n - 1 J- x 2 [c(n - 2)x n - 2 J 2c(n - l)xn - c(n - 2)xn = cnx n = an 149 (c) By part (b), C2nXn is one solution and by part (a) of Exercise 25 CIX n is another. Hence, by part (b) of Exercise 25, we conclude that CIX n + c2nx n is also a solution. Finally, two initial conditions will detennine Cl and C2 because they detennine two linear equations in the two unknowns Cl, C2 which are guaranteed to have a unique solution. 27. (a) We have Pn = rPn-l + SPn-2 + f(n) for n :2: 2 and also qn = rqn-l + sqn-2 for n :2: 2. Hence, rPn-l + SPn-2 + f(n) + rqn-l + sqn-2 r(Pn-l + qn-l) + S(Pn-2 + qn-2) + f(n) which says that Pn + qn satisfies the given relation for n :2: 2. (b) Part (a) shows that Pn + qn satisfies the given recurrence relation where qn = CIXl' + C2X~ or qn = CIX n + C2nXn according as the polynomial x 2 - rx - s has two distinct roots or just one. Two initial conditions provide two equations which can be solved for Cl, C2. Exercises 5.4 1. (a) [BB] Since (2 - 3x)2 = 4 - 12x + 9x 2 the associated sequence is
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