Special sequences: triangular, cube, geometric and Fibonacci-type

Not every sequence goes up by the same amount. Learn to spot the famous ones.

Practise this lesson free (6 questions)

Squares, cubes and triangles

Square numbers: 1, 4, 9, 16, 25 (n2)
Cube numbers: 1, 8, 27, 64, 125 (n3)
Triangular numbers: 1, 3, 6, 10, 15, ... add 2, then 3, then 4. The nth one is n(n + 1)2, so the 10th is 55.

Geometric sequences

Multiply by the same number each time, the common ratio r.
2, 6, 18, 54: r = 3. 81, 27, 9, 3: r = 13.
The ratio can be a surd (Higher): 1, √2, 2, 2√2, 4 has r = √2.

Fibonacci-type sequences

Each term is the sum of the two before it.
Starting 2, 5: 2, 5, 7, 12, 19
With letters: a, b, a + b, a + 2b, 2a + 3b. This lets you find unknown starting terms with equations.
A term-to-term rule says how to get the next term; a position-to-term rule (nth term) gives any term directly.
Worked example

A Fibonacci-type sequence has 3rd term 10 and 5th term 26. Find the first two terms.

  1. Terms: a, b, a + b, a + 2b, 2a + 3b
  2. a + b = 10 and 2a + 3b = 26
  3. Double the first: 2a + 2b = 20. Subtract: b = 6, so a = 4

Answer: 4 and 6

Key idea

Know square, cube and triangular numbers. Geometric: multiply by r. Fibonacci-type: add the previous two terms.

Key words
Ratio
A comparison of two or more quantities, written a : b.
Surd
A root that cannot be written exactly as a whole number or fraction, such as √5, left in exact form.
Term
A single number, letter or product of them in an expression, such as 5x or −3.
nth term
A rule that gives any term of a sequence from its position n, such as 3n + 2.

All GCSE and International GCSE Mathematics key terms and definitions →

Extra practice

2 more questions

Try each one, then tap to check.

  1. Find the 9th triangular number.

    Show the answer

    Answer: 45

    9 × 102 = 45.

  2. The sequence 4, 12, 36, 108, ... is geometric. What is the next term?

    Show the answer

    Answer: 324

    Each term is multiplied by 3: 108 × 3 = 324.

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Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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