Record a few cases before naming the rule.
When a sequence or growing shape appears, write down the position and the total beside it. Look first at the differences between totals. If the differences are themselves changing, record those too. This slows down an attractive but unreliable first guess.
Ask: what stays the same, what is added each time, and does the position number matter? A diagram can help when the quantities come from a shape; a table is usually clearer when they come from a number pattern.
Original worked example: growing tiles.
Pattern 1 uses 4 tiles. Pattern 2 uses 7 tiles. Pattern 3 uses 10 tiles. How many tiles does Pattern 8 use?
Make a table: each new pattern adds 3 tiles. From Pattern 1 to Pattern 8 there are 7 increases, so add 7 × 3 = 21 to 4. Pattern 8 uses 25 tiles.
A quick check is to continue: Pattern 4 would be 13 and Pattern 5 would be 16. The same increase of 3 is still visible, so 25 is consistent.
The rule can be written as “start with 4 and add 3 for every move after Pattern 1”. Writing that sentence makes the reasoning easier to inspect than jumping straight to a formula.
Self-check: test the rule before trusting it.
Try this fresh example: a sequence begins 6, 10, 14, 18. What is the 10th term?
Before reading on, write the increase and count how many increases occur between the first and tenth terms.
Check your reasoning
The increase is 4. There are 9 increases from term 1 to term 10, so 6 + (9 × 4) = 42. If you used ten increases, you counted the first term twice.
Use patterns as one option, then explain the choice.
In a mixed set, a pattern may be numerical, visual or hidden in a repeating cycle. Say what evidence made you choose a table or diagram. If no regular change appears after a few cases, stop forcing a pattern and try a different representation.
When the question tells you the final value and asks for an earlier one, working backwards may be the better method. After any question, use the solution-explanation structure to make the method reusable.
This article uses original examples. It is independent preparation advice, not official Australian Mathematics Competition material.