Unit 3
Repeating Pattern Sequence
About this unit
Identify the repeating cycle or growth rule in number, shape, and symbol sequences — then use it to find any term, no matter how far along the sequence. This unit also covers growing patterns where you need to find a formula for the nth figure.
What types of questions will you face?
- 1Find the element at a large position (e.g. 999th, 1004th) in a repeating symbol or number cycle
- 2Calculate how many elements (cubes, circles, matchsticks) will be in figure N of a growing pattern
- 3Identify the rule operating in an arithmetic or geometric number sequence
- 4Work out the position of a specific element within a pattern diagram
- 5Decode rule-based sequences where the rule toggles or transforms a state
Skills you will build
- Finding the cycle length of a repeating sequence
- Using division with remainder (modular arithmetic) to locate the nth term
- Identifying and extending linear and quadratic growing patterns
- Recognising Fibonacci, triangular, and other classic sequences
- Translating a visual pattern into a numerical formula
By the end of this unit, you will be able to
- Find the element at any position in a repeating sequence in seconds using remainders
- Generate a formula for how many elements will be in the nth figure of a growing pattern
- Recognise common sequence types and apply the right rule immediately
- Decode complex toggle-based or rule-based state sequences
Difficulty profile
Ranges from Very Easy (simple repeating cycles) to Difficult (complex growing patterns with combined rules). The Fibonacci sequence and toggle-based puzzles are the hardest questions in this unit.
Exam tip: Repeating Pattern Sequence
For repeating sequences: count the cycle length (e.g. 8 symbols), divide the position by that length, and the remainder tells you which symbol it is. If remainder = 0, it's the last in the cycle.
Sample Questions
Repeating-pattern questions often start with a simple table: spot how the numbers change from row to row, then predict the missing term.
Number-sequence and “what comes next?” items appear regularly on Selective TS — often as an approachable early question if you write the rule clearly.
The examiner wants you to identify the repeating or additive rule in the sequence (not just the last jump) and apply it to find the blank cell.
A grid or list shows several terms with one missing value. The pattern may repeat every few steps or grow by a fixed amount each time.
Best approach: List the differences between consecutive terms (or between rows). When the same step repeats, extend the pattern one more time and check which option matches.
Question
Given the number pattern, what is the missing number?
- A258
- B264
- C270
- D282
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Harder pattern items hide the rule in a story — you must translate the description into a sequence and see which option continues it correctly.
Rule-based sequence problems sit in the medium band on Selective papers; they reward careful reading and testing each answer against every stated constraint.
You must infer the placement or counting rule from the scenario, build the sequence mentally (or on paper), and pick the option that fits all rules — not just the last step.
Objects are arranged or counted under explicit rules (e.g. how many of each type, what repeats, what is forbidden). You choose the next valid state or total.
Best approach: Write the rule in your own words first. Generate the next one or two terms step by step. Eliminate any option that breaks a stated rule before selecting the survivor.
Question
An explorer finds an ancient altar with four glowing gems: Fire, Frost, Lightning, and Thunder. In front of the altar are four stone pedestals representing the seasons: Spring, Summer, Autumn, and Winter.
Touching a pedestal "toggles" the state of the gems it governs (turning a glowing gem dark, or a dark gem into a glow). Initially, all four gems are dark.
The explorer finds these inscriptions on the wall:
| Clues: |
|---|
| - The Spring pedestal awakens the Fire and Frost gems. |
| - The Summer pedestal affects the Lightning and Thunder gems. |
| - The Autumn pedestal changes the Fire, Frost, and Lightning gems. |
| - The Winter pedestal governs only the Thunder gem. |
Which of the following is a logical certainty regarding the state of the gems?
- AIf the Thunder gem is glowing, the Lightning gem must also be glowing.
- BIt is impossible to have only the Lightning gem glowing.
- CThe Fire gem and the Frost gem will always be in the same state.
- DTouching all four pedestals exactly once will result in all four gems glowing.
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