Patterns • Topic 3 of 4

Pattern Rules

A pattern rule is the sentence that captures how a pattern works, and there are two ways to say it. The recursive rule tells you how to get from one term to the next -- 'start at 5 and add 5 each time' for 5, 10, 15, 20. This is how children naturally describe a pattern, and it is enough to extend a sequence by a few steps. The functional rule (or position rule) links the term directly to its position number, so you can jump to any term without listing the ones before it: for 3, 6, 9, 12 the rule is 'term = 3 x position', meaning the 10th term is 3 x 10 = 30 without writing out the first nine. Knowing both matters because CTET sometimes asks for a distant term -- the 20th or 50th -- where adding one step at a time is too slow. A growing pattern usually hides addition or multiplication; a shrinking one hides subtraction or division. Pedagogically this is the doorway to algebra: when a child says 'add 4 each time' they are already doing the work that a formula like T = 4n will later make precise, which is why teachers are encouraged to have children put rules into their own words before any symbols appear.

✅ Solved examples

1. State the recursive rule for 4, 8, 12, 16, 20.
Start at 4 and add 4 to each term to get the next one. (In position form this is T = 4 x position, but the recursive 'add 4' is the simplest correct statement.)
2. The pattern is 2, 4, 6, 8, ... Using the rule 'term = 2 x position', what is the 9th term?
Apply the functional rule directly: term = 2 x 9 = 18. There is no need to list all nine terms.
3. What is the rule for 64, 32, 16, 8, ___, and what comes next?
Each term is half the previous one (the rule is 'divide by 2', a shrinking pattern). The next term is 8 / 2 = 4.
4. A pattern follows 'start at 7, add 5 each time'. Write the first four terms.
Begin at 7, then add 5 repeatedly: 7, 12, 17, 22.

✏️ Practice — try these, take hints as needed

1. State the recursive rule for 9, 18, 27, 36.
How do you get from one term to the next?
The jump is constant.
Start at 9 and add 9 each time
2. Using the rule 'term = 5 x position', find the 7th term.
Substitute the position straight into the rule.
No need to list earlier terms.
35
3. What is the rule for 90, 81, 72, 63?
The list is shrinking by the same amount.
Find the constant difference.
Subtract 9 from the previous term
4. A pattern follows 'start at 3, multiply by 2 each time'. Write the first four terms.
Double each term to get the next.
3, 6, 12, 24

📝 Topic test — 8 questions

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