The problem provides t1 = 5 and a recurrence t_n = 2 t_{n-1} - 8. The goal is to find t_n for a given n, and the provided numeric answer -3064 corresponds to a specific n. First, rewrite the recurrence to isolate the nonhomogeneous part: t_n - 8 = 2(t_{n-1} - 8). This shows the sequence u_n = t_n - 8 satisfies a simple homogeneous recurrence u_n = 2 u_{n-1}. Compute the initial condition for u_n: u_1 = t1 - 8 = 5 - 8 = -3. By unfolding the homogeneous recurrence, u_n = u_1 * 2^{n-1} = (-3) * 2^{n-1}. Translate back to t_n: t_n = u_n + 8 = 8 + (-3) * 2^{n-1} = 8 - 3 * 2^{n-1}. Now verify the given numeric value: suppose t_n = -3064. Then 8 - 3 * 2^{n-1} = -3064 => -3 * 2^{n-1} = -3072 => 2^{n-1} = 1024 = 2^{10} => n - 1 = 10 => n = 11. Thus, t_{11} = -3064, which matches the provided answer. If the question intended a general expression for t_n, the result is t_n = 8 - 3 * 2^{n-1} for all n ≥ 1. Note on the given options: the prompt shows options labeled a., b., c., d. but no content for the actual choices. In this case, we derived the correct closed form and the specific n that yields -3064, which aligns with the provided numeric result.

AI Solved: (K:1) | Recurrence relations | Mathematics | Black Tom

The Perrin sequence is a series of numbers where:P(1) = 3, P(2) = 0, P(3) = 2 P(n) = P(n-2) + P(n-3) for n > 3 The sequence begins: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ...Write a Python program to find a given number in the Perrin sequence.Use this to determine what is the sum of the 40th number and the 47th number in the Perrin sequence?Verification:4th Perrin number = 3 7th Perrin number = 5 Sum of 4th and 7th = 8 You may use the following for this question.Python InterpretersOnline Python Interpreter - online editor (onlinegdb.com) Python Tutor: Learn Python, JavaScript, C, C++, and Java programming by visualizing codeOnline Python Compiler (Interpreter) (programiz.com)Python ResourcesLearn Python Programming (programiz.com) Python Tutorial (w3schools.com)3.11.7 Documentation (python.org)https://www.pythoncheatsheet.org/

Given the following recursively defined sequence:f_1 = 1, f_n = (\sqrt{f_{n-1}} + 1)^2Which of the following is a correct non-recursive definition for the same sequence?

Consider the sequence defined by f_n = 2^{n}, which begins 1, 2, 4, 8, 16, 32...Which of the following is a correct recursive definition of this sequence?

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Given the following recursively defined sequence:f_1 = 1, f_n = (\sqrt{f_{n-1}} + 1)^2Which of the following is a correct non-recursive definition for the same sequence?

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