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 x . x f ( x ) = x a 0 + x a 1 x + x a 2 x 2 + ........... + x a n x n + ......... = a 0 x + a 1 x 2 + a 2 x 3 + ........... + a n x n + 1 + ......... Therefore, f ( x ) − x f ( x ) = a 0 + a 1 x + a 2 x 2 + ........ a n x n + ..... − ( x a 0 + x 2 a 1 + x 3 a 2 + ....... x n a n − 1 + x n + 1 a n + ...... ) = a 0 + ( a 1 − a 0 ) x + ( a 2 − a 1 ) x 2 + ..... + ( a n − a n − 1 ) x n + ....... It is given that a n = a n − 1 + 2 n for n ≥ 2 , so a n − a n − 1 = 2 n . Also for n = 1 , a 1 − a 0 = 2 . For, n = 2 , a 2 − a 1 = 2 2 = 4 . Continuing like this ( a n − a n − 1 ) = 2 n . Thus, f ( x ) − x f ( x ) = a 0 + ( a 1 − a 0 ) x + ( a 2 − a 1 ) x 2 +

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 5.4, Problem 7E

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The formula for an given a0=5 and an=an−1+2n for n≥1 using the generating functions.

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Chapter 5.4, Problem 6E

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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 formula for a n given a 0 = 5 and a n = a n − 1 + 2 n for n ≥ 1 using the generating functions.

Solution Summary: The author explains how to determine the formula for a_n using the generating functions.

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The formula for an given a0=5 and an=an−1+2n for n≥1 using the generating functions.

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