Number Patterns

How to spot special sequences and use pattern recognition

By Mr Gan · O-Level E Maths · Updated August 2026 · 8 min read

A special sequence is one that follows a known family, such as square numbers 1, 4, 9, 16, ..., cube numbers 1, 8, 27, 64, ..., triangular numbers 1, 3, 6, 10, ..., or the Fibonacci-style rule where each term is the sum of the two before it. You spot the family first by checking the values against these patterns, then by looking at first and second differences. Once you match the family, the nth term follows a fixed formula rather than needing to be built from scratch.

Why students freeze on these questions

Most sequence questions in class are linear (constant difference) or quadratic (constant second difference), and students get comfortable finding the nth term for those. Special sequences look unfamiliar because the difference pattern is not constant and not a simple second difference either, so the standard nth term routine does not immediately apply. The real skill is recognition: knowing the handful of standard families on sight so you are matching a pattern, not inventing a formula under time pressure.

The good news is that O-Level E-Math only tests a small, fixed set of special sequences. Learn what each one looks like and its nth term, and every question becomes a matching exercise.

The method: name the family, then apply its nth term

Step 1

Write out the terms and their position number (1st, 2nd, 3rd, ...)

Step 2

Check against known families: squares, cubes, triangular, powers

Step 3

If no family matches by sight, find 1st then 2nd differences

Step 4

Apply the matched family's nth term, check against 2 terms

Watch the whole method in about a minute.

The standard families to memorise

Square numbers

1, 4, 9, 16, 25, ...
nth term n²

Cube numbers

1, 8, 27, 64, 125, ...
nth term n³

Triangular numbers

1, 3, 6, 10, 15, ...
nth term n(n + 1) ÷ 2

Powers of 2

2, 4, 8, 16, 32, ...
nth term 2ⁿ

Fibonacci-style

1, 1, 2, 3, 5, 8, ...
add the two terms before

Each family has a giveaway. Triangular numbers are built by adding one more each time (add 2, then 3, then 4), so their differences climb by 1. Powers of 2 double at every step, which is why they sit so close to the index laws. Fibonacci-style sequences are defined by their rule rather than by a formula: at O-Level you are asked to state the rule and continue the sequence, not to produce a closed nth term.

Worked example 1: triangular numbers

Find the next two terms and the nth term of 1, 3, 6, 10, 15, ....

Solution

Find the first differences: 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4, 15 − 10 = 5. The differences themselves go up by 1 each time, so this is not a constant-difference (linear) sequence.

Differences that climb by 1 point to the triangular family, but they do not settle it on their own: triangular numbers with a constant added have exactly the same differences. Check the values themselves. Here 1, 3, 6, 10, 15 are the standard triangular numbers, so no adjustment is needed.

The next difference after 5 is 6, so the 6th term is 15 + 6 = 21. The next difference after that is 7, so the 7th term is 21 + 7 = 28.

Apply the triangular number formula, nth term = n(n + 1) ÷ 2. Check n = 4: 4 × 5 ÷ 2 = 10 ✓. Check n = 5: 5 × 6 ÷ 2 = 15 ✓.

Quick check: triangular numbers are the running totals of 1, 2, 3, 4, 5, ... (1, then 1+2=3, then 1+2+3=6, and so on). Differences of exactly 2, 3, 4, 5, ... tell you the sequence is triangular or triangular shifted by a constant, so compare one actual term against n(n + 1) ÷ 2 before you commit to the nth term.

Worked example 2: a sequence that mixes squares with a shift

Find the nth term of 2, 5, 10, 17, 26, ....

Solution

First differences: 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7, 26 − 17 = 9. Not constant, so try second differences.

Second differences: 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2. Constant second difference of 2 means this is a quadratic sequence, and the coefficient of n² is half the second difference: 2 ÷ 2 = 1, so the n² term is n².

Compare each term to n²: n = 1 gives 1² = 1, but the actual term is 2, a difference of +1. n = 2 gives 2² = 4, actual term 5, again +1. The pattern holds for all terms.

So the nth term is n² + 1. Check n = 5: 5² + 1 = 26 ✓.

The step students get wrong

Spotting "these look like square numbers" is not enough on its own: 2, 5, 10, 17, 26 is not the square sequence itself, it is squares shifted by a constant. Always subtract your guessed pattern (n², n³, or the triangular formula) from the actual terms to find the leftover constant or linear adjustment, then check that adjustment against at least two terms before committing to an answer.

Worked example 3: Fibonacci-style rule

A sequence starts 3, 5, 8, 13, 21, .... Describe the rule and find the next term.

Solution

Check whether each term is the sum of the two before it: 3 + 5 = 8 ✓, 5 + 8 = 13 ✓, 8 + 13 = 21 ✓.

This confirms a Fibonacci-style rule: each term equals the sum of the previous two terms.

The next term is 13 + 21 = 34.

Exam tip: for Fibonacci-style sequences, O-Level questions usually ask you to state the rule in words ("each term is the sum of the two terms before it") and continue the sequence, not to write a closed nth term formula. Do not spend time hunting for one that does not exist in the syllabus.

Frequently asked questions

How do I know if a sequence is linear, quadratic, or one of the special types?

Check first differences: if they are constant, it is linear, and you can use the standard method in our nth term of a sequence guide. If first differences are not constant but second differences are, it is quadratic. If neither is constant, check the actual term values against the special families (squares, cubes, triangular numbers) before assuming the sequence is unusual.

Do I need to memorise the cube number sequence?

Yes, cube numbers 1, 8, 27, 64, 125 (that is, 1³ to 5³) come up often enough that recognising them instantly saves time. If a sequence's differences grow quickly and irregularly, checking against n³ is worth trying before working through differences by hand.

What if a sequence looks like a mix of two patterns?

This is common, as in worked example 2 where the sequence was square numbers plus a constant. Always try matching the closest standard family first (squares, cubes, or triangular numbers), then check what needs to be added or subtracted to make it fit exactly.

Is pattern recognition tested with diagrams, not just number lists?

Yes, many O-Level questions show a sequence of dot patterns or matchstick shapes and ask for the number of dots or sticks in the nth pattern. Count the actual quantities in the first three or four diagrams to turn it into a number sequence, then apply the same method: check known families, then differences.

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