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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BlackTom AI · Accepted Answer

To evaluate the correct recursive definition, I’ll examine each option in turn and relate it to the given explicit formula f_n = 2^n and its initial term.
Option a: 'f_1 = 1, f_n = f_{n-1} × 2'. This matches the sequence because starting from f_1 = 1 and doubling at each step yields f_2 = 2, f_3 = 4, f_4 = 8, and so on, which precisely aligns with 2^n for n ≥ 1. The recursive rule uses the previous term to generate the next term, which is the standard form for a geometric progression on the sequence values.
Option b: 'f_1 = 1, f_n = 2^{n-1} × 2'. Although this evaluates to f_n = 2^n, the expression depends on n directly rather than on the previous term f_{n-1}. A recursive definition should express f_n in terms of earlier terms (and possibly constants), not in terms of n in a way that bypasses the previous term; moreover, this form isn’t a true recursion since it doesn’t define f_n solely via f_{n-1} or earlier terms.
Option c: 'f_1 = 1, f_n = 2^{f_{n-1}} × 2'. This would produce a radically different growth pattern because the exponent depends on f_{n-1}, not on n, leading to astronomically faster growth than 2^n. It does not reproduce the given sequence and is not a valid recursive definition for f_n = 2^n.
Option d: 'f_1 = 1, f_n = (n-1) × 2'. This is a linear formula in n, yielding 0, 2, 4, 6, ... after n = 1, which clearly does not match the exponential sequence 1, 2, 4, 8, 16, ...; hence it fails to define the same terms.
In summary, only option a provides a recursive rule that generates the same sequence from the initial term by repeatedly doubling, while the other options either fail to use the previous term, produce a different growth pattern, or do not reproduce 2^n at each step.

FIT1058_S1_2025 Week 6 Pre-reading Quiz

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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