Explanation It is given that a 1 = 1 . Now, substitute k = 1 in a k + 1 = k 2 a k a 2 = 1 2 a 1 = 1.1 = 1 For k = 2 , a 3 = k 2 a 2 = 2 2 .1 = 4 For k = 3 , a 4 = k 2 a 3 = 3 2 .4 = 9.4 = 36 For k = 4 , a 5 = k 2 a 4 = 4 2 .36 = 16.36 = 576 For k = 5 , a 6 = k 2 a 5 = 5 2 .576 = 25.576 = 14400 Thus, the first six terms of the sequence are 1,1,4,36,576,14400. On observing the values calculated in part (a), we get a n = ( ( n − 1 ) ! ) 2 for n ≥ 1 . We will use the principle of mathematical induction to prove a n = ( ( n − 1 ) ! ) 2 for n ≥ 1 . For n = 1 , a 1 = ( ( 1 − 1 ) ! ) 2 = ( 1 ) 2 = 1 , which is true

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 14E

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Find the first six terms of the sequence a1,a2,a3,....... with a1=1, and for k≥1, ak+1=k2ak.

Guess a formula for an and prove that your guess is correct.

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Chapter 5.2, Problem 13E

Chapter 5.2, Problem 15E

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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 six terms of the sequence a 1 , a 2 , a 3 , ....... with a 1 = 1 , and for k ≥ 1 , a k + 1 = k 2 a k . Guess a formula for a n and prove that your guess is correct.

Solution Summary: The author explains how to determine the first six terms of the sequence with a_1, kge 1, and a formula, using the principle of mathematical induction.

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Find the first six terms of the sequence a1,a2,a3,....... with a1=1, and for k≥1, ak+1=k2ak.

Guess a formula for an and prove that your guess is correct.

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Chapter 5 Solutions

Find the first six terms of the sequence a 1 , a 2 , a 3 , ....... with a 1 = 1 , and for k ≥ 1 , a k + 1 = k 2 a k . Guess a formula for a n and prove that your guess is correct.

Solution Summary: The author explains how to determine the first six terms of the sequence with a_1, kge 1, and a formula, using the principle of mathematical induction.

Concept explainers

Videos

Find the first six terms of the sequence a1,a2,a3,....... with a1=1, and for k≥1, ak+1=k2ak. Guess a formula for an and prove that your guess is correct.

Want to see the full answer?

Chapter 5 Solutions

Find the first six terms of the sequence a1,a2,a3,....... with a1=1, and for k≥1, ak+1=k2ak.

Guess a formula for an and prove that your guess is correct.