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Consequently, equation (7.47) takes the following form: k k (-1)k 1+ V5 F-k = V5 - or F_k = (-1)k+1Fg.

Consequently, equation (7.47) takes the following form: k k (-1)k 1+ V5 F-k = V5 - or F_k = (-1)k+1Fg.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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explain the determine purple 

This number, called the golden mean, represents the ratio of the sides of a rectangle that is supposed to be “most pleasing" to the eye. Consider now the values of the Fibonacci numbers for the index k being a negative integer. The most direct way of obtaining these values is to substitute -k for k in equation (7.42). Doing this gives -k 1 + V5 V5 F-k V5 (7.47) k k 2 2 V5 + V5, V5 However, 2 2 1 1 – V5 (7.48) 1+ V5 1+ V5 1 2 and 2 1+ V5 (7.49) 1 – V5 220 Difference Equations Consequently, equation (7.47) takes the following form: k k V5 (-1)* V5 + V5 F-k (7.50) - 2 2 or F-k = (-1)k+1Fr. (7.51) k• This is the required relationship between the Fibonacci numbers for negative and positive values of the index k. As a final example of the properties of the Fibonacci numbers, we use the principle of mathematical induction to prove the relation F1 + F2 + + Fk Fk+2 – 1. (7.52)
Transcribed Image Text:This number, called the golden mean, represents the ratio of the sides of a rectangle that is supposed to be “most pleasing" to the eye. Consider now the values of the Fibonacci numbers for the index k being a negative integer. The most direct way of obtaining these values is to substitute -k for k in equation (7.42). Doing this gives -k 1 + V5 V5 F-k V5 (7.47) k k 2 2 V5 + V5, V5 However, 2 2 1 1 – V5 (7.48) 1+ V5 1+ V5 1 2 and 2 1+ V5 (7.49) 1 – V5 220 Difference Equations Consequently, equation (7.47) takes the following form: k k V5 (-1)* V5 + V5 F-k (7.50) - 2 2 or F-k = (-1)k+1Fr. (7.51) k• This is the required relationship between the Fibonacci numbers for negative and positive values of the index k. As a final example of the properties of the Fibonacci numbers, we use the principle of mathematical induction to prove the relation F1 + F2 + + Fk Fk+2 – 1. (7.52)
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