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(a) Let (F(n))n>o be a sequence of real numbers and let k e N. Explain what it means for (F(n))n>0 to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n20 satisfies the following recurrence relation: F(n) — аF(n — 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n>0. (c) If (F(n)n>o is a sequence, write down the generating function for (F(n))n>0-

(a) Let (F(n))n>o be a sequence of real numbers and let k e N. Explain what it means for (F(n))n>0 to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n20 satisfies the following recurrence relation: F(n) — аF(n — 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n>0. (c) If (F(n)n>o is a sequence, write down the generating function for (F(n))n>0-

Advanced Engineering Mathematics
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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2. (a) Let (F(n)n>o be a sequence of real numbers and let k E N. Explain what it means for (F(n))n>o to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n>0 satisfies the following recurrence relation: F(n) = aF(n – 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n > 0. (c) If (F(n))n>0 is a sequence, write down the generating function for (F(n))n>0- (d) A sequence (F(n))n>0 has the following initial terms F(0) = 2, F(1) = 4, F(2) = 10 and satisfies the recurrence relation F(n) = 3F(n – 1) – 2 for n > 1. 1 (i) Show that this sequence also satisfies the relation F(n) = 4F(n – 1) – 3F(n – 2) for n> 2. (ii) Using the generating functions approach, find a closed form for F(n) for all n> 0.
Transcribed Image Text:2. (a) Let (F(n)n>o be a sequence of real numbers and let k E N. Explain what it means for (F(n))n>o to satisfy a (k + 1)-term recurrence relation with constant coefficients. (b) Suppose (F(n))n>0 satisfies the following recurrence relation: F(n) = aF(n – 1) for n>1 and F(0) = b, where a, b e N. Compute the values of F(1), F(2), and then find a general formula for F(n) for arbitrary n > 0. (c) If (F(n))n>0 is a sequence, write down the generating function for (F(n))n>0- (d) A sequence (F(n))n>0 has the following initial terms F(0) = 2, F(1) = 4, F(2) = 10 and satisfies the recurrence relation F(n) = 3F(n – 1) – 2 for n > 1. 1 (i) Show that this sequence also satisfies the relation F(n) = 4F(n – 1) – 3F(n – 2) for n> 2. (ii) Using the generating functions approach, find a closed form for F(n) for all n> 0.
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