Fibonacci-type sequences

Objective. Explore recurrence relations and invariant patterns.

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Learn: the key idea

A Fibonacci-type sequence is a recurrence relation where each term is the sum of the two terms before it, un+2 = un+1 + un, requiring two starting values rather than one. The classic Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, each term found by adding the previous two. These sequences can show invariant patterns, such as the ratio of consecutive terms approaching the golden ratio as the sequence grows. To extend such a sequence, always add the two most recent terms, not the two smallest or the first and last. Check your extension against the given terms before continuing further.

Worked example

A Fibonacci-type sequence begins 2, 5, 7, 12, 19, … Find the next two terms.

1 $Check rule: each term = sum of previous two, e.g. 7+12=19 ✓$
2 $Next term = 12 + 19 = 31$
3 $Following term = 19 + 31 = 50$
4 $Sequence continues 2, 5, 7, 12, 19, 31, 50$

Answer: 31, 50

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Common mistake. Pupils add only the single previous term instead of the two previous terms, treating it like a normal recursive rule. For 1, 1, 2, 3, 5 they might compute the next term as 5+1=6 using the wrong pair instead of 5+3=8. Always add the two most recent consecutive terms.

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