Spotting what a pattern is doing — a constant difference, a constant ratio, or something else entirely — and extending it, including patterns given as diagrams or tables, with worked examples, a worksheet with memo and a quiz.
A pattern (or sequence) is a list of numbers that follows a rule. Each number in the list is called a term. The whole skill is working out what the pattern is doing — and there are only a few possibilities to check.
1Step 1: always find the differences first
Write the gaps between consecutive terms underneath the pattern. If those gaps are all the same, you have a constant difference — the most common type in Grade 8.
- 1Find the differences: 12 − 7 = 5, 17 − 12 = 5, 22 − 17 = 5.Always subtract a term from the one after it.
- 2The difference is constant: +5 each time.This makes it a linear (constant-difference) pattern.
- 3Next three terms: 22 + 5 = 27, then 32, then 37.Keep adding the same difference.
2Decreasing patterns
If the terms get smaller, the difference is simply negative. Nothing else changes.
- 1Differences: 90 − 100 = −10, and the same each time.A negative difference means a decreasing pattern.
- 2Next three terms: 60, 50, 40.Keep subtracting 10.
3Constant ratio (geometric) patterns
If the differences are not constant, check whether each term is multiplied by the same number. Divide a term by the one before it — if you always get the same answer, that number is the ratio.
- 1Differences are 4, 12, 36 — not constant, so try dividing.This is the signal to test for a ratio.
- 26 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3.The ratio is constant: ×3 each time.
- 3Next three terms: 162, 486, 1 458.Keep multiplying by 3.
Quick test: if a pattern grows slowly and steadily it is usually a constant difference; if it explodes (2, 6, 18, 54 …) it is usually a constant ratio.
4Patterns that are neither
The ATP is clear that Grade 8 patterns are not limited to constant differences and ratios. If neither works, look for a familiar family or a changing difference.
- Square numbers: 1, 4, 9, 16, 25, … (n × n)
- Cube numbers: 1, 8, 27, 64, …
- Triangular numbers: 1, 3, 6, 10, 15, … (the difference grows by 1 each time)
- Add the two before it: 1, 1, 2, 3, 5, 8, 13, …
- 1Differences: 2, 3, 4 — not constant, but they increase by 1 each time.A changing-but-predictable difference is still a pattern.
- 2The next difference must be 5.
- 3Next terms: 10 + 5 = 15, then +6 → 21, then +7 → 28.These are the triangular numbers.
5Patterns given as diagrams
Exams often show a pattern built from matchsticks, dots or tiles. Count the items in each picture, write those counts as a number pattern, and then work exactly as before.
- 1Write the counts as a pattern: 4, 7, 10, …Turn the pictures into numbers first.
- 2The difference is 3 (each new square only needs 3 more sticks — one side is shared).Looking at the picture explains the number.
- 34th picture: 10 + 3 = 13 matchsticks.
Do not guess from the first two terms alone. 2, 4, … could be +2 (2, 4, 6, 8) or ×2 (2, 4, 8, 16). Check every gap you are given before deciding — a third term settles it: 2, 4, 6 is +2 and 2, 4, 8 is ×2.
6Two ways to describe a rule
Grade 8 asks you to describe a rule in your own words, and there are two kinds. A term-to-term rule says how to get from one term to the next. A position-to-term rule says how to get any term straight from its position, without listing the terms before it.
- 1Term to term: start at 3 and add 4 each time.
- 2Position to term: position 1 gives 3, position 2 gives 7, position 3 gives 11. Each term is 4 × position − 1.Check: 4 × 1 − 1 = 3 and 4 × 2 − 1 = 7.
- 3The 20th term is 4 × 20 − 1 = 79.The term-to-term rule would need 19 additions to get there.
- 1The terms go up by 5, so the rule starts with 5 × position.
- 25 × 1 = 5, but the first term is 6, so add 1: term = 5 × position + 1.Check: 5 × 4 + 1 = 21.
- 3Position 10: 5 × 10 + 1 = 51.
A term-to-term rule is quick for the next few terms. A position-to-term rule is the sensible way to find a term far down the list, such as the 100th.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Write the next three terms: 6, 11, 16, 21, …
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Difference +5 → 26, 31, 36 - 2Write the next three terms: 45, 38, 31, 24, …
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Difference −7 → 17, 10, 3 - 3Write the next three terms: 3, 9, 27, 81, …
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Ratio ×3 → 243, 729, 2 187 - 4Write the next three terms: 160, 80, 40, 20, …
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Ratio ÷2 → 10, 5, 2,5 - 5Write the next two terms: 1, 4, 9, 16, 25, …
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Square numbers → 36, 49 - 6Write the next two terms: 1, 3, 6, 10, 15, …
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Triangular; differences 2,3,4,5,6… → 21, 28 - 7Write the next two terms: 1, 1, 2, 3, 5, 8, …
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Add the two before → 13, 21 - 8A pattern of tiles gives 5, 9, 13, … How many tiles in the 5th shape?
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Difference +4 → 5, 9, 13, 17, 21 tiles - 9Is 2, 6, 18, 54 a constant-difference or constant-ratio pattern? Give the value.
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Constant ratio, ×3 (differences 4, 12, 36 are not constant)
Quick Quiz
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