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?

Question

BlackTom AI · Accepted Answer

Consider the given recursive sequence: f1 = 1 and f_n = (sqrt(f_{n-1}) + 1)^2.

Option a: f_n = n^2. Start from n = 1: f1 = 1 = 1^2, which matches. Assume f_{n-1} = (n-1)^2 for some n > 1. Then sqrt(f_{n-1}) = sqrt((n-1)^2) = n-1 (taking the positive root since f_{n-1} is a square). Compute f_n = ( (n-1) + 1 )^2 = n^2. This matches the proposed non-recursive form, and the induction step holds, so f_n = n^2 is consistent with the recurrence for all n ≥ 1.

Option b: f_n = (n+1)^2. If we test n = 2 using the recurrence, f2 = (sqrt(f1) + 1)^2 = (1 + 1)^2 = 4, but (n+1)^2 with n = 2 gives 9. This contradicts the actual value from the recurrence, so this form is not correct.

Option c: f_n = sqrt(n). For n = 1, sqrt(1) = 1, which matches f1, but for n = 2 we would get sqrt(2), whereas the recurrence yields f2 = 4. Since these diverge at the next term, this cannot be the correct non-recursive definition.

Option d: f_n = sqrt(n+1). For n = 1, sqrt(2) ≈ 1.414, which does not equal f1 = 1, so this option fails immediately at the base case and cannot describe the sequence.

In summary, checking the base case and the induction step shows that only the form f_n = n^2 is consistent with the recurrence for all n ≥ 1, while the other expressions fail either at the base case or at a subsequent step.

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?单项选择题

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