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Consider the following recurrence relation and initial conditions. tk  =  8tk − 1 − 16tk − 2, for each integer k ≥ 2 t0  =  1,   t1 = 4 (a) Suppose a sequence of the form 1, t, t2, t3,   , tn   , where t ≠ 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation?

Consider the following recurrence relation and initial conditions. tk  =  8tk − 1 − 16tk − 2, for each integer k ≥ 2 t0  =  1,   t1 = 4 (a) Suppose a sequence of the form 1, t, t2, t3,   , tn   , where t ≠ 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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Consider the following recurrence relation and initial conditions.
tk  =  8tk − 1 − 16tk − 2, for each integer k ≥ 2
t0  =  1,   t1 = 4
(a)
Suppose a sequence of the form 1, t, t2, t3,   , tn   , where t ≠ 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation?
Consider the following recurrence relation and initial conditions. tk = 8t -1- 16t, - 2, for each integer k 2 2 to = 1, t, = 4 (a) Suppose a sequence of the form 1, t, t², t3,..., t" where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation? What value of t is solution to this equation? t = (b) Suppose a sequence to, t,, t,, ... satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for to, t,, t,, ... in terms of n. It follows from part (a) and the single roots theorem that for some constants C and D, the terms of to, t,, t,, ... satisfy the equation t, = for every integer n 2 0. Solve for C and D by setting up a system of two equations in two unknowns using the facts that t, = 1 and t, = 4. The result is that t, = for every integer n 2 0.
Transcribed Image Text:Consider the following recurrence relation and initial conditions. tk = 8t -1- 16t, - 2, for each integer k 2 2 to = 1, t, = 4 (a) Suppose a sequence of the form 1, t, t², t3,..., t" where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation? What value of t is solution to this equation? t = (b) Suppose a sequence to, t,, t,, ... satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for to, t,, t,, ... in terms of n. It follows from part (a) and the single roots theorem that for some constants C and D, the terms of to, t,, t,, ... satisfy the equation t, = for every integer n 2 0. Solve for C and D by setting up a system of two equations in two unknowns using the facts that t, = 1 and t, = 4. The result is that t, = for every integer n 2 0.
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