Explanation Let f ( x ) = a 0 + a 1 x + a 2 x 2 + ........ a n x n + ..... be the generating function for the sequence. Multiplying both sides by 2 x . 2 x f ( x ) = 2 x a 0 + 2 x a 1 x + 2 x a 2 x 2 + ........... + 2 x a n x n + ......... = 2 a 0 x + 2 a 1 x 2 + 2 a 2 x 3 + ........... + 2 a n x n + 1 + ......... Multiplying both sides by x 2 . x 2 f ( x ) = x 2 a 0 + x 2 a 1 x + x 2 a 2 x 2 + ........... + x 2 a n x n + ......... = x 2 a 0 + a 1 x 3 + a 2 x 4 + ........... + a n x n + 2 + ......... Multiplying both sides by 2 x 3 . 2 x 3 f ( x ) = 2 x 3 a 0 + 2 x 3 a 1 x + 2 x 3 a 2 x 2 + ........2 x 3 a n x n + ..... = 2 x 3 a 0 + 2 a 1 x 4 + 2 a 2 x 5 + ........2 a n x n + 3 + ..... Therefore, f ( x ) − 2 x f ( x ) − x 2 f ( x ) + 2 x 3 f ( x ) = ( a 0 + a 1 x + a 2 x 2 + ........ a n x n + ..... ) − ( 2 a 0 x + 2 a 1 x 2 + 2 a 2 x 3 + ........... + 2 a n x n + 1 + ......... ) − ( x 2 a 0 + a 1 x 3 + a 2 x 4 + ........... + a n x n + 2 + ......... ) + ( 2 x 3 a 0 + 2 a 1 x 4 + 2 a 2 x 5 + ........2 a n x n + 3 + ..... ) ⇒ f ( x ) − 2 x f ( x ) − x 2 f ( x ) + 2 x 3 f ( x ) = a 0 + ( a 1 − 2 a 0 ) x + ( a 2 − 2 a 1 − a 0 ) x 2 + ( a 3 − 2 a 2 − a 1 + 2 a 0 ) x 3 ........ ( a n − 2 a n − 1 − a n − 2 + 2 a n − 3 ) x n + ..... It is given that a n = 2 a n − 1 + a n − 2 − 2 a n − 3 for n ≥ 3 , so a n − 2 a n − 1 − a n − 2 + 2 a n − 3 = 0 . So, for n = 3 , a 3 − 2 a 2 − a 1 + 2 a 0 = 0 . For n = 4 , a 4 − 2 a 3 − a 2 + 2 a 1 = 0 . Similarly the following terms will be 0

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

3rd Edition

ISBN: 9780134689555

Author: Edgar Goodaire, Michael Parmenter

Publisher: PEARSON

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Chapter 5.4, Problem 11E

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The solution of the recurrence relation an=2an−1+an−2−2an−3,n≥3, given a0=1,a1=3,a2=6.

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Chapter 5.4, Problem 10E

Chapter 5.4, Problem 12E

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Chapter 5 Solutions

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 5.1 - True/False Questions The statement i=1n(2i1)=n2...Ch. 5.1 - Prob. 2TFQ Ch. 5.1 - Prob. 3TFQ Ch. 5.1 - Prob. 4TFQ Ch. 5.1 - Prob. 5TFQ Ch. 5.1 - Prob. 6TFQ Ch. 5.1 - Prob. 7TFQ Ch. 5.1 - Prob. 8TFQ Ch. 5.1 - Prob. 9TFQ Ch. 5.1 - Prob. 10TFQ

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The solution of the recurrence relation a n = 2 a n − 1 + a n − 2 − 2 a n − 3 , n ≥ 3 , given a 0 = 1 , a 1 = 3 , a 2 = 6.

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The solution of the recurrence relation an=2an−1+an−2−2an−3,n≥3, given a0=1,a1=3,a2=6.

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Chapter 5 Solutions