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Number Patterns: Types, the nth Term and Examples
Published: 12.08.2026·Updated: 14.08.2026
Nadia Rahmawati
Math Curriculum Team, Algonova
A number pattern is a sequence of numbers arranged according to a rule. Each term can be obtained from the previous one by applying that same rule. It is taught in Year 8 and leads directly into sequences and series at senior level — and into the algorithmic thinking used in programming.
Types of Number Pattern
| Pattern | Example | nth term |
|---|---|---|
| Odd numbers | 1, 3, 5, 7, 9 | Un = 2n − 1 |
| Even numbers | 2, 4, 6, 8, 10 | Un = 2n |
| Square | 1, 4, 9, 16, 25 | Un = n² |
| Rectangular | 2, 6, 12, 20, 30 | Un = n(n + 1) |
| Triangular | 1, 3, 6, 10, 15 | Un = n(n + 1) ÷ 2 |
| Fibonacci | 1, 1, 2, 3, 5, 8, 13 | Un = Un−1 + Un−2 |
| Pascal (row totals) | 1, 2, 4, 8, 16 | Un = 2ⁿ⁻¹ |
How to Find the Rule
This is the core skill, and the order never changes:
Step 1 — find the differences between consecutive terms.
For 3, 7, 11, 15 the differences are 4, 4, 4. A constant difference means an arithmetic pattern: Un = 4n − 1.
Step 2 — if the difference is not constant, look for a ratio.
For 2, 6, 18, 54 each term is multiplied by 3. This is geometric: Un = 2 × 3ⁿ⁻¹.
Step 3 — if neither is constant, take the difference of the differences.
For 1, 4, 9, 16 the differences are 3, 5, 7, and the second differences are 2, 2. A constant second difference means the pattern involves n².
Step 4 — check whether each term is the sum of the two before it. If so, it is Fibonacci.
The nth Term of an Arithmetic Sequence
For patterns with a constant difference:
where a is the first term and b is the common difference.
Example: For 5, 8, 11, 14, ... what is the 20th term?
a = 5, b = 3 → U₂₀ = 5 + (20−1) × 3 = 5 + 57 = 62
That is exactly why the formula is taught: writing out the sequence to reach term 20 takes twenty steps, the formula takes one.
A Worked Example
Consider the pattern 2, 6, 12, 20, 30, ... Find the 10th term.
The differences are 4, 6, 8, 10 — not constant. The second differences are 2, 2, 2 — constant, so the pattern involves n². Notice that 2 = 1×2, 6 = 2×3, 12 = 3×4, 20 = 4×5. So Un = n(n+1).
U₁₀ = 10 × 11 = 110
The Mistakes That Actually Cost Marks
1. Guessing from only two terms. Two numbers fit infinitely many rules. Check at least three differences before concluding anything.
2. Miscounting n. The 20th term uses (n−1) = 19, not 20. This produces answers that are out by exactly one common difference — an error pattern that is easy to recognise once you know it.
3. Stopping after step one. When the difference is not constant many children give up, yet steps two to four are what exam questions actually test.
4. Confusing difference with ratio. A difference is added, a ratio is multiplied. The sequence 2, 6, 18 has ratio 3, not difference 4.
5. Memorising formulas without their origin. A child who sees that the rectangular pattern is n(n+1) because the shape really is an n by (n+1) rectangle will never forget the formula.
