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Let (F(n))n>0 be a sequence of real numbers and let k e N. (a) Explain what it means for (F(n))n>o to satisfy a (k +1)-term recurrence relation with constant coefficients. (b) Write down the generating function for (F(n))n>0- (c) Suppose the sequence (F(n))n>0 has the following initial terms F(0) = 1, F(1) = 4, and satisfies the recurrence relation F(n) = 2F(n – 1) + 8F(n – 2) for n>2.

Let (F(n))n>0 be a sequence of real numbers and let k e N. (a) Explain what it means for (F(n))n>o to satisfy a (k +1)-term recurrence relation with constant coefficients. (b) Write down the generating function for (F(n))n>0- (c) Suppose the sequence (F(n))n>0 has the following initial terms F(0) = 1, F(1) = 4, and satisfies the recurrence relation F(n) = 2F(n – 1) + 8F(n – 2) for n>2.

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 56EQ
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2. Let (F(n))n>o be a sequence of real numbers and let k € N. (a) Explain what it means for (F(n))n>0 to satisfy a (k +1)-term recurrence relation with constant coefficients. (b) Write down the generating function for (F(n))n>0- (c) Suppose the sequence (F(n))n>o has the following initial terms F(0) = 1, F(1) = 4, and satisfies the recurrence relation F(n) = 2F(n – 1) + 8F(n – 2) for n> 2. | Using any approach, find a closed form for F(n) for all n > 0.
Transcribed Image Text:2. Let (F(n))n>o be a sequence of real numbers and let k € N. (a) Explain what it means for (F(n))n>0 to satisfy a (k +1)-term recurrence relation with constant coefficients. (b) Write down the generating function for (F(n))n>0- (c) Suppose the sequence (F(n))n>o has the following initial terms F(0) = 1, F(1) = 4, and satisfies the recurrence relation F(n) = 2F(n – 1) + 8F(n – 2) for n> 2. | Using any approach, find a closed form for F(n) for all n > 0.
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