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A sequence eo ₁, 2 satisfies the following recurrence relation and initial conditions. ek = gek-2 for each integer k 2 2. €1=2 0, Find an explicit formula for the sequence.

A sequence eo ₁, 2 satisfies the following recurrence relation and initial conditions. ek = gek-2 for each integer k 2 2. €1=2 0, Find an explicit formula for the sequence.

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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A sequence eo, e₁,e2, satisfies the following recurrence relation and initial conditions. ek = gek-2 for each integer k 2 2. eo 0, - e₁ = 2 Find an explicit formula for the sequence. As a first step, find the roots of the characteristic equation for the recurrence relation. If 1, t, t², t3,t, satisfies the recurrence relation for t * 0, then e = tk = Dividing both sides by tk-2 shows that t² = Thus, the characteristic equation is and its roots are t = 3 and t = = 0, Hence, by the ---Select-- .([ en = C.3" +D. ... Since e = 0 and e₁ = 2, then or, equivalently, |= C. 3⁰ + D. ( =C+D = 3C - |= C.3¹ +D. (1 ([ or, equivalently, en ✓theorem there are constants C and D such that the terms of eo, e₁,e₂, satisfy the equation for every integer n 2 0. 8 :-(¯)p. When these two simultaneous equations are solved, the result is C= and D = 2. Thus, an explicit formula for eo, e₁,e₂, is en --?--✓ 0 for every integer n 20. • 3² + (¯)•(¯)* +(-1)-1.3 |--?-V *** if n is odd if n is even. for each integer k 2 0. Thus, by definition of eo, e₁,e₂, X
Transcribed Image Text:A sequence eo, e₁,e2, satisfies the following recurrence relation and initial conditions. ek = gek-2 for each integer k 2 2. eo 0, - e₁ = 2 Find an explicit formula for the sequence. As a first step, find the roots of the characteristic equation for the recurrence relation. If 1, t, t², t3,t, satisfies the recurrence relation for t * 0, then e = tk = Dividing both sides by tk-2 shows that t² = Thus, the characteristic equation is and its roots are t = 3 and t = = 0, Hence, by the ---Select-- .([ en = C.3" +D. ... Since e = 0 and e₁ = 2, then or, equivalently, |= C. 3⁰ + D. ( =C+D = 3C - |= C.3¹ +D. (1 ([ or, equivalently, en ✓theorem there are constants C and D such that the terms of eo, e₁,e₂, satisfy the equation for every integer n 2 0. 8 :-(¯)p. When these two simultaneous equations are solved, the result is C= and D = 2. Thus, an explicit formula for eo, e₁,e₂, is en --?--✓ 0 for every integer n 20. • 3² + (¯)•(¯)* +(-1)-1.3 |--?-V *** if n is odd if n is even. for each integer k 2 0. Thus, by definition of eo, e₁,e₂, X
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