(a) 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 5 x . 5 x f ( x ) = 5 x a 0 + 5 x a 1 x + 5 x a 2 x 2 + ........... + 5 x a n x n + ......... = 5 a 0 x + 5 a 1 x 2 + 5 a 2 x 3 + ........... + 5 a n x n + 1 + ......... Therefore, f ( x ) + 5 x f ( x ) = a 0 + a 1 x + a 2 x 2 + ........ a n x n + ..... + ( 5 x a 0 + 5 x 2 a 1 + 5 x 3 a 2 + .......5 x n a n − 1 + 5 x n + 1 a n + ...... ) = a 0 + ( a 1 + 5 a 0 ) x + ( a 2 + 5 a 1 ) x 2 + ..... + ( a n + 5 a n − 1 ) x n + ....... It is given that a n = − 5 a n − 1 for n ≥ 1 , so a n + 5 a n − 1 = 0 . Also for n = 1 , a 1 + 5 a 0 = 0 . For n = 2 , a 2 + 5 a 1 = 0 . Similarly the following terms will be 0. Thus, f ( x ) + 5 x f ( x ) = a 0 + ( a 1 + 5 a 0 ) x + ( a 2 + 5 a 1 ) x 2 + (b) To determine The solution of a n = − 5 a n − 1 + 3 , n ≥ 1 , given a 0 = 2 using generating functions. Now, use the methods of Section 5.3 to verify your answer.

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 9E

(a)

To determine

The solution of an=−5an−1,n≥1, given a0=2 using generating functions. Now, use the methods of Section 5.3 to verify your answer.

(b)

To determine

The solution of an=−5an−1+3,n≥1, given a0=2 using generating functions. Now, use the methods of Section 5.3 to verify your answer.

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Chapter 5.4, Problem 8E

Chapter 5.4, Problem 10E

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The solution of a n = − 5 a n − 1 , n ≥ 1 , given a 0 = 2 using generating functions. Now, use the methods of Section 5.3 to verify your answer.