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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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 57EQ
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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
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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