Explanation It is given that a 1 = 0 and a 2 = 1 2 . Now, substitute k = 1 in a k + 2 = 1 2 ( a k + a k + 1 ) a 3 = 1 2 ( a 1 + a 2 ) = 1 2 ( 0 + 1 2 ) = 1 4 For k = 2 , a 4 = 1 2 ( a 2 + a 3 ) = 1 2 ( 1 2 + 1 4 ) = 1 2 ( 2 + 1 4 ) = 3 8 For k = 3 , a 5 = 1 2 ( a 3 + a 4 ) = 1 2 ( 1 4 + 3 8 ) = 1 2 ( 2 + 3 8 ) = 5 16 For k = 4 , a 6 = 1 2 ( a 4 + a 5 ) = 1 2 ( 3 8 + 5 16 ) = 1 2 ( 6 + 5 16 ) = 11 32 For k = 5 , a 7 = 1 2 ( a 5 + a 6 ) = 1 2 ( 5 16 + 11 32 ) = 1 2 ( 10 + 11 32 ) = 21 64 Thus, the first seven terms of the sequence are 0 , 1 2 , 1 4 , 3 8 , 5 16 , 11 32 , 21 64 . We will use the principle of mathematical induction to prove a n = 1 3 ( 1 − ( − 1 2 ) n − 1 ) for all n ≥ 1 . For n = 1 , a 1 = 1 3 ( 1 − ( − 1 2 ) 1 − 1 ) = 1 3 ( 1 − 1 ) = 0 . Thus, the given formula is true for n = 1 . Let us assume that the given formula is true for n = k . That is, a k = 1 3 ( 1 − ( − 1 2 ) k − 1 ) for all k ≥ 1 . Now, we need to prove that the formula is true for n = k + 1 . Thus, we need to prove that a k + 1 = 1 3 ( 1 − ( − 1 2 ) k + 1 − 1 ) = 1 3 ( 1 − ( − 1 2 ) k )

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 5.2, Problem 5E

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Find the first seven terms of the sequence a1,a2,a3,....... with a1=0, a2=12 and ak+2=12(ak+ak+1) for k≥1.

Also, prove that an=13(1−(−12)n−1) for every n≥1.

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Chapter 5.2, Problem 4E

Chapter 5.2, 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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Find the first seven terms of the sequence a 1 , a 2 , a 3 , ....... with a 1 = 0 , a 2 = 1 2 and a k + 2 = 1 2 ( a k + a k + 1 ) for k ≥ 1 . Also, prove that a n = 1 3 ( 1 − ( − 1 2 ) n − 1 ) for every n ≥ 1 .

Solution Summary: The author explains how to determine the first seven terms of the sequence a_1, a..2,mathrm.......

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Find the first seven terms of the sequence a1,a2,a3,....... with a1=0, a2=12 and ak+2=12(ak+ak+1) for k≥1.

Also, prove that an=13(1−(−12)n−1) for every n≥1.

Want to see the full answer?

Chapter 5 Solutions

Find the first seven terms of the sequence a 1 , a 2 , a 3 , ....... with a 1 = 0 , a 2 = 1 2 and a k + 2 = 1 2 ( a k + a k + 1 ) for k ≥ 1 . Also, prove that a n = 1 3 ( 1 − ( − 1 2 ) n − 1 ) for every n ≥ 1 .

Solution Summary: The author explains how to determine the first seven terms of the sequence a_1, a..2,mathrm.......

Concept explainers

Videos

Find the first seven terms of the sequence a1,a2,a3,....... with a1=0, a2=12 and ak+2=12(ak+ak+1) for k≥1. Also, prove that an=13(1−(−12)n−1) for every n≥1.

Want to see the full answer?

Chapter 5 Solutions

Find the first seven terms of the sequence a1,a2,a3,....... with a1=0, a2=12 and ak+2=12(ak+ak+1) for k≥1.

Also, prove that an=13(1−(−12)n−1) for every n≥1.