Explanation Given information: It turns out that the Fibonacci sequence satisfy the following explicit formula: For all integer F n ≥ 0 , F n = 1 5 [ ( 1 + 5 2 ) n + 1 − ( 1 − 5 2 ) n + 1 ] Proof: Let k be an integer such that k ≥ 2 . Let a = 1 + 5 2 and let b = 1 − 5 2 , which we will use to simplify the used expressions. F n = 1 5 ⋅ [ a n + 1 − b n + 1 ] We then obtain: F k − 1 + F k − 2 = 1 5 ⋅ [ a k − b k ] + 1 5 ⋅ [ a k − 1 − b k − 1 ] = 1 5 ⋅ [ a k − b k + a k − 1 − b k − 1 ] [ Distributive property ] = 1 5 ⋅ [ ( a k + a k − 1 ) − ( b k − 1 + b k ) ] Let us first evaluate a k + a k − 1 : a k + a k − 1 = a ⋅ a k − 1 + 1 ⋅ a k − 1 = ( a + 1 ) ⋅ a k − 1 [ Distributed property ] = ( 1 + 5 2 + 1 ) ⋅ a k − 1 [ ∵ a = 1 + 5 2 ] = 3 + 5 2 ⋅ a k − 1 = 6 + 2 5 4 ⋅ a k − 1 = 1 + 2 5 + 5 2 2 ⋅ a k − 1 = 1 2 + 2 5 + ( 5 ) 2 2 2 ⋅ a k − 1 = ( 1 + 5 ) 2 2 2 ⋅ a k − 1 [ Use ( a + b ) 2 = a 2 + 2 a b + b 2 ] = ( 1 + 5 2 ) 2 ⋅ a k − 1 = a 2 ⋅ a k − 1 [ ∵ a = 1 + 5 2 ] = a 2 +

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ISBN: 9780357540244

Author: EPP

Publisher: CENGAGE L

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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 5.6, Problem 33ES

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To prove:

For all integers k≥2, Prove that the sequence defined by the given formula satisfy the recurrence relation Fk=Fk−1+Fk−2.

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Chapter 5.6, Problem 32ES

Chapter 5.6, Problem 34ES

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Chapter 5 Solutions

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To prove: For all integers k ≥ 2 , Prove that the sequence defined by the given formula satisfy the recurrence relation F k = F k − 1 + F k − 2 .

Solution Summary: The author demonstrates that the Fibonacci sequence satisfy the recurrence relation for all integers kge 2.

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To prove:

For all integers k≥2, Prove that the sequence defined by the given formula satisfy the recurrence relation Fk=Fk−1+Fk−2.

Want to see the full answer?

Chapter 5 Solutions

To prove: For all integers k ≥ 2 , Prove that the sequence defined by the given formula satisfy the recurrence relation F k = F k − 1 + F k − 2 .

Solution Summary: The author demonstrates that the Fibonacci sequence satisfy the recurrence relation for all integers kge 2.

Concept explainers

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To prove: For all integers k≥2, Prove that the sequence defined by the given formula satisfy the recurrence relation Fk=Fk−1+Fk−2.

Want to see the full answer?

Chapter 5 Solutions

To prove:

For all integers k≥2, Prove that the sequence defined by the given formula satisfy the recurrence relation Fk=Fk−1+Fk−2.