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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57a26049-260e-45aa-910f-dd852150673b%2Fe5c5eade-6b77-40e3-a34a-e67c358078c9%2Fqfnemat_processed.jpeg&w=3840&q=75)
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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