Recognise and Work with Fibonacci-Type Sequences

Algebra • Sequences

Flashcards

Test your knowledge with interactive flashcards

Does the sequence 1, 2, 3, 6 fit Fibonacci-type?

Click to reveal answer

1 + 2 = 3 holds, but 2 + 3 = 5 does not equal 6, so the sequence does not fit Fibonacci-type.

Key concepts

What you'll likely be quizzed about

Definition of a Fibonacci-type sequence

A Fibonacci-type sequence is a sequence (a_n) defined by a recurrence relation a_n = a_{n-1} + a_{n-2} for all n ≥ 3. The sequence requires two initial terms a_1 and a_2; different initial values produce different sequences. The recurrence relation causes each later term to depend only on the two immediately preceding terms.

Recognising Fibonacci-type sequences

Recognition starts by checking that each term from the third term onward equals the sum of the two previous terms. If the check holds for consecutive terms, the sequence fits the Fibonacci-type recurrence. A single failure in the check proves that the sequence is not Fibonacci-type under that recurrence.

Generating terms from starting values

Given a_1 and a_2, the recurrence a_n = a_{n-1} + a_{n-2} generates all later terms deterministically. The initial choice causes growth behaviour: positive starting values cause terms to grow quickly, while sign changes or zeros produce different patterns. The recurrence prevents independent choice of later terms once the first two are fixed.

Worked example and verification

Example: a_1 = 2, a_2 = 3 gives a_3 = 5, a_4 = 8, a_5 = 13. Verification proceeds by calculating a_3 and comparing with the listed third term. If the listed term matches the calculated value, the recurrence holds for that position. Repeated verification across the sequence establishes that the sequence is Fibonacci-type.

Limiting factors and special cases

The recurrence a_n = a_{n-1} + a_{n-2} requires two initial terms; sequences missing either initial term cannot be fully determined. The sequence behaviour depends on initial values: zeros or negative numbers change sign patterns, and non-integer starting values produce non-integer terms. The recurrence holds only when explicitly stated; other linear recurrences, such as a_n = 2a_{n-1} + a_{n-2}, are not Fibonacci-type.

Key notes

Important points to keep in mind

A Fibonacci-type sequence requires two initial terms and the recurrence a_n = a_{n-1} + a_{n-2}.

Verification requires checking that each term from the third onward equals the sum of the two previous terms.

Forward generation uses a_3 = a_2 + a_1, then continues iteratively to produce later terms.

Backward reconstruction uses a_{n-2} = a_n − a_{n-1} and can recover earlier terms when later consecutive terms are known.

Single or partial matches to the recurrence do not prove the whole sequence is Fibonacci-type; check all provided terms.

Zero or negative starting values are valid but change term behaviour; non-integer starts produce non-integer terms.

Recurrence coefficients must be 1 and 1; any other coefficients define a different linear recurrence.