1. Linear recurrence relation(You will likely see this topic in a course on discrete math.)The Fibonacci sequence, named after the Italian mathematician Leonard Fibonacci of Pisa,is the most well-known sequence of numbers. A lot of the numbers in this sequence appearin nature (see the video Nature by Numbers: The Golden Ratio and Fibonacci Numbers)The first two terms of the Fibonacci sequence arefi = 1and f2 = 1.The subsequent terms of the Fibonacci sequence are given by the recurrence relationfnfn-1+ fn-2,for n > 3.||Here are the first few terms of the Fibonacci sequence:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...Let A =Observe thatAA?= Aand A3In general, we have|fn-1[fn-2[fn-1]= Aand= An-2for n > 3.fnfn-1]fn1This matrix A has two eigenvalues 1tv5 and y with eigenvectors1and-V5 :1+/52respectively. The diagonalization of A is PDP-l where11 -1+V511+v5P =P-1andD =21+/51-V5 )V51+V51-V52 As we will see in Section 7.3.1,A? = (PDP-1)(PDP 1)= PD(P-1P)DP-1 = PDIDP-1 = PD²P¬1andAk = PD*P-1for k > 1.As a result, we get the numbers in the Fibonacci sequence as follows:fn-1An-2fnPDn-2p-1п-211 =1+v5112-1+/51-V5()n-2V51+v5n-1п-1-1+/5/5Võ[ ()" - (4)We conclude that, for n > 1, the n-th term of the Fibonacci sequence isn1+V51fnV51-2Note that y5 is called the golden ratio.Now consider the sequence defined by M1 = 0, M2 = 1 andМn 3 2Mп-1 + 3Mп-2,for n > 3.(a) Write down the first five terms of the sequence.(b) Write the matrix A satisfying|Mn-1|Mn-2= AMnfor n > 3.Mn-1(c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.(d) Use the fact that Ak = PD*P-lto get a formula for Mn.

Note:- As per our guidelines, we can answer the first three sub-parts of this problem. Please…

%3D Now consider the sequence defined by M1 = 0, M2 Mn = 2Mn-1 +3Mn-2, for n > 3. (a) Write down the first five terms of the sequence. (b) Write the matrix A satisfying [Ma-2] = A Mn-1 [Mn-1 for n > 3. Mn (c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.

%3D Now consider the sequence defined by M1 = 0, M2 Mn = 2Mn-1 +3Mn-2, for n > 3. (a) Write down the first five terms of the sequence. (b) Write the matrix A satisfying [Ma-2] = A Mn-1 [Mn-1 for n > 3. Mn (c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.

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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Question

1. Linear recurrence relation
(You will likely see this topic in a course on discrete math.)
The Fibonacci sequence, named after the Italian mathematician Leonard Fibonacci of Pisa,
is the most well-known sequence of numbers. A lot of the numbers in this sequence appear
in nature (see the video Nature by Numbers: The Golden Ratio and Fibonacci Numbers)
The first two terms of the Fibonacci sequence are
fi = 1
and f2 = 1.
The subsequent terms of the Fibonacci sequence are given by the recurrence relation
fn
fn-1+ fn-2,
for n > 3.
||
Here are the first few terms of the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Let A =
Observe that
A
A?
= A
and A3
In general, we have
|fn-1
[fn-2
[fn-1]
= A
and
= An-2
for n > 3.
fn
fn-1]
fn
1
This matrix A has two eigenvalues 1tv5 and y with eigenvectors
1
and
-V5 :
1+/5
2
respectively. The diagonalization of A is PDP-l where
1
1 -1+V5
1
1+v5
P =
P-1
and
D =
2
1+/5
1-V5 )
V5
1+V5
1-V5
2

As we will see in Section 7.3.1,
A? = (PDP-1)(PDP 1)= PD(P-1P)DP-1 = PDIDP-1 = PD²P¬1
and
Ak = PD*P-1
for k > 1.
As a result, we get the numbers in the Fibonacci sequence as follows:
fn-1
An-2
fn
PDn-2p-1
п-2
1
1 =1+v5
1
1
2
-
1+/5
1-V5
()
n-2
V5
1+v5
n-1
п-1-
1+/5
/5
Võ[ ()" - (4)
We conclude that, for n > 1, the n-th term of the Fibonacci sequence is
n
1+V5
1
fn
V5
1-
2
Note that y5 is called the golden ratio.
Now consider the sequence defined by M1 = 0, M2 = 1 and
Мn 3 2Mп-1 + 3Mп-2,
for n > 3.
(a) Write down the first five terms of the sequence.
(b) Write the matrix A satisfying
|Mn-1
|Mn-2
= A
Mn
for n > 3.
Mn-1
(c) Diagonalize A, that is, write A = PDP-1 for some diagonal matrix D.
(d) Use the fact that Ak = PD*P-lto get a formula for Mn.

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