6. The sequence 1, 1, 3, 7, 17, 41, 99,... is a second-order linear recurrence given by x₂ = 2x₁+x_₂ for n ≥2 with x₁ =1 & x₁ = 1 X₂ i) If we write the recurrence as H ]-[ :} what is the 2 x 2 matrix A? Xe-1 X₂ ii) If we let then = A¹, for some power & that is a function of n. X-l What is k as a function of n, precisely? iii) Find a diagonal form for A, that is, find invertible matrix P and diagonal matrix D such that A = PDP-¹. iv) Use the information above to find an explicit formula for x, in terms of n. (Test for various values of n to make sure it works. X-1
6. The sequence 1, 1, 3, 7, 17, 41, 99,... is a second-order linear recurrence given by x₂ = 2x₁+x_₂ for n ≥2 with x₁ =1 & x₁ = 1 X₂ i) If we write the recurrence as H ]-[ :} what is the 2 x 2 matrix A? Xe-1 X₂ ii) If we let then = A¹, for some power & that is a function of n. X-l What is k as a function of n, precisely? iii) Find a diagonal form for A, that is, find invertible matrix P and diagonal matrix D such that A = PDP-¹. iv) Use the information above to find an explicit formula for x, in terms of n. (Test for various values of n to make sure it works. X-1
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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![6. The sequence 1, 1, 3, 7, 17, 41, 99,... is a second-order linear recurrence given by
x₁ = 2x₁ + x2 for n≥2 with x = 1 & x₁ = 1
X₁
i)
If we write the recurrence as
H
what is the 2 x 2 matrix A?
X-2
X₁
ii)
If we let
=
[2] the
then = A* for some power k that is a function of n.
What is k as a function of n, precisely?
iii)
Find a diagonal form for A, that is, find invertible matrix P and diagonal matrix
D such that A = PDP-¹.
iv)
Use the information above to find an explicit formula for x in terms of n. (Test
for various values of n to make sure it works.
اسپرتیکر
X₁-1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F543d5259-ea45-47e4-b2f7-40deeb6480b7%2F33b77410-a678-4e8e-b12e-2ccdd95586aa%2F6ospkeo_processed.png&w=3840&q=75)
Transcribed Image Text:6. The sequence 1, 1, 3, 7, 17, 41, 99,... is a second-order linear recurrence given by
x₁ = 2x₁ + x2 for n≥2 with x = 1 & x₁ = 1
X₁
i)
If we write the recurrence as
H
what is the 2 x 2 matrix A?
X-2
X₁
ii)
If we let
=
[2] the
then = A* for some power k that is a function of n.
What is k as a function of n, precisely?
iii)
Find a diagonal form for A, that is, find invertible matrix P and diagonal matrix
D such that A = PDP-¹.
iv)
Use the information above to find an explicit formula for x in terms of n. (Test
for various values of n to make sure it works.
اسپرتیکر
X₁-1
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