Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence is Recursively the Fibonacci sequence is ao = 0, a₁ = 1, an+1 = an+an-1 for n ≥ 1. In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix equation is true: (a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we have: @=[ JQ-E J(J·D=J · J (J). = (b) Let A = Following a similar argument, show that This generalizes! For all n ≥ 1, we have that 1-610 0, 1, 1, 2, 3, 5, 8, 11, 19,,.... 1- and [¹]=[][](Equation 1) an-1 (Equation 2) You can take Equation 2 for granted, though it's a direct generalization of the work you've done! [J √5 2 Av=Xv.). [an+1] an sic -eigvenctor is 5 B-JO = Explain why this is enough to conclude that A is diagonalizable. 1+ √5] = (c) Let's denote the eigenvalues in part (b) as + and o- 2 number + is called the golden ratio and is one of the most bers in mathematics! Show that a basic +-eigenvector is = Find CA (x) and verify that that the two eigenvalues of A are [1] 1+ √5] 2 1- √5 The 2 famous num- and that a ba- (you don't need to solve systems, just use the equation
Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence is Recursively the Fibonacci sequence is ao = 0, a₁ = 1, an+1 = an+an-1 for n ≥ 1. In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix equation is true: (a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we have: @=[ JQ-E J(J·D=J · J (J). = (b) Let A = Following a similar argument, show that This generalizes! For all n ≥ 1, we have that 1-610 0, 1, 1, 2, 3, 5, 8, 11, 19,,.... 1- and [¹]=[][](Equation 1) an-1 (Equation 2) You can take Equation 2 for granted, though it's a direct generalization of the work you've done! [J √5 2 Av=Xv.). [an+1] an sic -eigvenctor is 5 B-JO = Explain why this is enough to conclude that A is diagonalizable. 1+ √5] = (c) Let's denote the eigenvalues in part (b) as + and o- 2 number + is called the golden ratio and is one of the most bers in mathematics! Show that a basic +-eigenvector is = Find CA (x) and verify that that the two eigenvalues of A are [1] 1+ √5] 2 1- √5 The 2 famous num- and that a ba- (you don't need to solve systems, just use the equation
Chapter9: Sequences, Probability And Counting Theory
Section9.2: Arithmetic Sequences
Problem 4SE: What are the main differences between using a recursive formula and using an explicit formula to...
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![2. Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of
numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence
is
Recursively the Fibonacci sequence is
an+1 = an+an-1 for n ≥ 1.
In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using
eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix
equation is true:
ao
-
(b) Let A
and
0,
=
a 1
0, 1, 1, 2, 3, 5, 8, 11, 19,, ....
An+1
an
(Equation 1)
(a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we
have:
sic -eigvenctor is
Av = Xv.).
(d) Write the vector
@= @= J (J·D=[JEJ (JD).
1
B
=1,
=
Following a similar argument, show that
This generalizes! For all n ≥ 1, we have that
=
3
an+1
An
[-]
an
an-1_
You can take Equation 2 for granted, though it's a direct generalization of the work
you've done!
1 + √5
J Find CÃ (x) and verify that that the two eigenvalues of A are 2
Explain why this is enough to conclude that A is diagonalizable.
1+ √5]
1- √5
(c) Let's denote the eigenvalues in part (b) as 0+
and o- =
2
2
number + is called the golden ratio and is one of the most famous num-
bers in mathematics! Show that a basic +-eigenvector is
and that a ba-
The
1
1J
1-6 JO
=
N
(Equation 2)
as a linear combination of
=
(you don't need to solve systems, just use the equation
[+]
and
(e) Use Equation 2, part (b) and part (d) to show that an
formula!
[4]
=
on - on
√5
What a wild](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F293ec77a-bb0f-4498-a317-90c98547c742%2F386b18a5-b198-4895-bd4e-63823eede9c3%2F98070pu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of
numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence
is
Recursively the Fibonacci sequence is
an+1 = an+an-1 for n ≥ 1.
In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using
eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix
equation is true:
ao
-
(b) Let A
and
0,
=
a 1
0, 1, 1, 2, 3, 5, 8, 11, 19,, ....
An+1
an
(Equation 1)
(a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we
have:
sic -eigvenctor is
Av = Xv.).
(d) Write the vector
@= @= J (J·D=[JEJ (JD).
1
B
=1,
=
Following a similar argument, show that
This generalizes! For all n ≥ 1, we have that
=
3
an+1
An
[-]
an
an-1_
You can take Equation 2 for granted, though it's a direct generalization of the work
you've done!
1 + √5
J Find CÃ (x) and verify that that the two eigenvalues of A are 2
Explain why this is enough to conclude that A is diagonalizable.
1+ √5]
1- √5
(c) Let's denote the eigenvalues in part (b) as 0+
and o- =
2
2
number + is called the golden ratio and is one of the most famous num-
bers in mathematics! Show that a basic +-eigenvector is
and that a ba-
The
1
1J
1-6 JO
=
N
(Equation 2)
as a linear combination of
=
(you don't need to solve systems, just use the equation
[+]
and
(e) Use Equation 2, part (b) and part (d) to show that an
formula!
[4]
=
on - on
√5
What a wild
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