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.

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A.a.

B.b.

C.c.

D.d.

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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 ......Login to view full explanation

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