Which of the following gives a complete recursive definition of a sequence?

Question

BlackTom AI · Accepted Answer

To determine which option provides a complete recursive definition of a sequence, I will examine each choice and note what a complete recursive definition requires.
Option a: f_1 = 1, f_2 = 4, f_3 = 9, f_4 = 16, f_5 = 25.
- This lists several explicit values but does not provide a rule that generates future terms from previous ones. It is an explicit listing (not recursive) of the first few terms, so it is not a complete recursive definition.

Option b: f_n = f_{n-1} + 2.
- This recurrence tells us how to obtain f_n from f_{n-1}, but it lacks initial condition(s). A complete recursive definition must specify at least one base case (e.g., f_1 = ... or f_0 = ...). Without a starting value, the sequence is undefined for n beyond the base case. Therefore, this is incomplete as a recursive definition.

Option c: f_n = (2n-1 : n ∈ ℕ).
- This appears to be an explicit formula that describes each term directly in terms of n (the nth odd number), not a rule that generates terms from previous terms. It is explicit rather than recursive, so it does not meet the requirement for a complete recursive definition.

Option d: f_1 = 1, f_2 = 1, f_n = f_{n-1} + f_{n-2}.
- This provides both base cases (f_1 and f_2) and a recursive rule that uses previous terms to define the next one. This structure matches the standard form of a complete recursive definition, such as that of the Fibonacci sequence, and indeed fully specifies how to construct the entire sequence.

Summary: Options a, b, and c either lack a base case, describe an explicit (nonrecursive) formula, or fail to provide a true recursive rule. Option d includes both the necessary base conditions and a recursive generation rule, making it a complete recursive definition.

FIT1058_S1_2025 Week 6 Pre-reading Quiz

Which of the following gives a complete recursive definition of a sequence?

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