Fibonacci Sequence and Fibonacci-Type Sequences
A Fibonacci sequence starts 1, 1 and makes each new term by adding the two terms before it. This gives 1, 1, 2, 3, 5 and so on. A Fibonacci-type sequence follows the same rule but can start with any two numbers.
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Spot the Fibonacci Family
- Add the last two terms to get the next term.
- The Fibonacci sequence starts with 1 and 1.
- A Fibonacci-type sequence can start with any two numbers.
The Classic Fibonacci Sequence
- The Fibonacci sequence starts with 1 and 1.
- Each new term is the sum of the two terms before it.
- , , and so on.
Fibonacci-Type Sequences
- A Fibonacci-type sequence follows the same adding rule.
- The difference is that it can start with any two numbers.
- The first two numbers determine the whole sequence.
Finding the Next Terms
- Add the first two terms to find the third term.
- Keep adding the last two terms to continue the sequence.
- , , , and so on.
Finding Missing Terms
- Work backwards when later terms are already given.
- Subtract the previous known term to find the missing term before it.
Practice Questions
Test your understanding
Find the next term in the Fibonacci-type sequence 4, 5, 9, 14, 23, ...
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In a Fibonacci-type sequence, each term is the sum of the two terms before it.
Check the rule: , and .
So add the last two terms: .
The next term is 37.
In a Fibonacci-type sequence, each term is the sum of the two terms before it. Which of these is a Fibonacci-type sequence?
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Check the rule on each option: every term should be the sum of the two before it.
For 3, 7, 10, 17, 27: , and .
The rule holds all the way through, so this is a Fibonacci-type sequence.
The other options grow by adding a fixed amount, by doubling, or only fit the rule for the first pair.
Find the missing term in the Fibonacci-type sequence 6, _, 15, 24, 39, ...
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Each term is the sum of the two before it, so the 3rd term equals the 1st term plus the missing 2nd term.
This gives , so the missing term is .
Check it works: , and .
A Fibonacci-type sequence starts 4, 6, where each term is the sum of the two before it. What is the 6th term?
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Each term is the sum of the two before it, so keep adding the last two terms and count the positions carefully.
1st: 4, 2nd: 6, 3rd: , 4th: , 5th: , 6th: .
The sequence is , so the 6th term is 42.
A Fibonacci-type sequence has 4th term 25 and 5th term 40. What is the 2nd term?
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The sequence so far is and we need to work backwards by subtracting.
The 5th term is the 3rd term plus the 4th term, so the 3rd term is .
The 4th term is the 2nd term plus the 3rd term, so the 2nd term is .
So the 2nd term is 10. (The full sequence is .)
A Fibonacci-type sequence has 5th term 31 and 6th term 51. What is the 1st term?
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Each term is the sum of the two before it, so work backwards by subtracting.
4th term: .
3rd term: .
2nd term: .
1st term: .
So the sequence is and the 1st term is 2.
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Interactive Activity
Find the missing or next term in these Fibonacci and Fibonacci-type sequences
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Frequently asked questions
The questions students bump into most on this topic
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on: it starts with 1 and 1, and every term after that is the sum of the two before it. 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5. The fifth Fibonacci number is 5. It is named after Leonardo Fibonacci, an Italian mathematician of the 1200s.
Add the last two terms to get the next one. After 1, 1, 2, 3, 5, 8, 13, the next term is 8 + 13 = 21, and the one after is 13 + 21 = 34. That single rule builds the whole sequence; you never need a formula, just the two terms you have most recently written.
A sequence that follows the Fibonacci rule, each term is the sum of the two before it, but starts with any two numbers rather than 1 and 1. 1, 5, 6, 11, 17 is one, and so is 3, 8, 11, 19, 30. The two starting numbers decide the whole sequence. GCSE questions usually give one of these and ask for terms.
Add the two most recent terms. Starting with 2 and 5: 2 + 5 = 7 is the third term, 5 + 7 = 12 the fourth, and 7 + 12 = 19 the fifth, giving 2, 5, 7, 12, 19. Keep going the same way for as many terms as the question asks.
Use the same rule backwards: subtract. If the 4th term is 22 and the 5th is 35, the 3rd is 35 − 22 = 13, the 2nd is 22 − 13 = 9 and the 1st is 13 − 9 = 4, so the sequence is 4, 9, 13, 22, 35. Each subtraction undoes one addition.
In school maths it is the classic example of a sequence defined by a rule rather than a formula, and GCSE papers use Fibonacci-type sequences to test that rule. Outside maths, Fibonacci numbers turn up in nature: the number of petals on many flowers, the spirals of seeds in a sunflower and the scales on a pine cone are often Fibonacci numbers.
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