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Consider the following recurrence relation and initial conditions. bk = 9bk - 1- 18b, - 2, for every integer k 2 2 bo = 2, b, = 4 (a) Suppose a sequence of the form 1, t, t2, t³, .., tª ., where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation? What are the possible values of t? (Enter your answer as a comma-separated list.) t = (b) Suppose a sequence bo, b,, b,, ... satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for bo, b,, b2, ... in terms of n. It follows from part (a) and the --Select--- v roots theorem that for some constants C and D, the terms of bo, b,, b,, ... satisfy the equation b, = fo 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 b, = 2 and b, = 4. The result is that b, = for every integer n 2 0.

Consider the following recurrence relation and initial conditions. bk = 9bk - 1- 18b, - 2, for every integer k 2 2 bo = 2, b, = 4 (a) Suppose a sequence of the form 1, t, t2, t³, .., tª ., where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the characteristic equation of the recurrence relation? What are the possible values of t? (Enter your answer as a comma-separated list.) t = (b) Suppose a sequence bo, b,, b,, ... satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for bo, b,, b2, ... in terms of n. It follows from part (a) and the --Select--- v roots theorem that for some constants C and D, the terms of bo, b,, b,, ... satisfy the equation b, = fo 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 b, = 2 and b, = 4. The result is that b, = for every integer n 2 0.

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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Consider the following recurrence relation and initial conditions. 9b k - 1 18bk - 2' for every integer k 2 2 b. 0, = 2, b1 = 4 (a) Suppose a sequence of the form 1, t, t2, t³, characteristic equation of the recurrence relation? ..., where t # 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the What are the possible values of t? (Enter your answer as a comma-separated list.) t = (b) Suppose a sequence bo, b1, bo, b1, b2, satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for in terms of n. It follows from part (a) and the ---Select--- v roots theorem that for some constants C and D, the terms of bo, b,, b,, satisfy the equation b, for 1' 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 b. = 2 and b, = 4. The result is that bn for every integer n 2 0.
Transcribed Image Text:Consider the following recurrence relation and initial conditions. 9b k - 1 18bk - 2' for every integer k 2 2 b. 0, = 2, b1 = 4 (a) Suppose a sequence of the form 1, t, t2, t³, characteristic equation of the recurrence relation? ..., where t # 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the What are the possible values of t? (Enter your answer as a comma-separated list.) t = (b) Suppose a sequence bo, b1, bo, b1, b2, satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for in terms of n. It follows from part (a) and the ---Select--- v roots theorem that for some constants C and D, the terms of bo, b,, b,, satisfy the equation b, for 1' 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 b. = 2 and b, = 4. The result is that bn for every integer n 2 0.
Consider the following recurrence relation and initial conditions. tk 12tk - 1 36tg for each integer k 2 2 %D 2' to = 1, t, = 6 (a) Suppose a sequence of the form 1, t, t², t³, characteristic equation of the recurrence relation? where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the What value of t is a solution to this equation? t = (b) Suppose a sequence to, t, t2 satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for in terms of n. It follows from part (a) and the --Select--- v 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 to = 1 and t, = 6. The result is that t, for every integer n 2 0.
Transcribed Image Text:Consider the following recurrence relation and initial conditions. tk 12tk - 1 36tg for each integer k 2 2 %D 2' to = 1, t, = 6 (a) Suppose a sequence of the form 1, t, t², t³, characteristic equation of the recurrence relation? where t + 0, satisfies the given recurrence relation (but not necessarily the initial conditions). What is the What value of t is a solution to this equation? t = (b) Suppose a sequence to, t, t2 satisfies the given initial conditions as well as the recurrence relation. Fill in the blanks below to derive an explicit formula for in terms of n. It follows from part (a) and the --Select--- v 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 to = 1 and t, = 6. The result is that t, for every integer n 2 0.
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