Understand the idea.
Two different kinds of rule
A term-to-term rule explains the step from one value to the next. To generate a sequence from it, you also need the first value. Starting at five and adding three gives , but adding three alone does not specify the start.
A term-number rule uses the position directly. Multiplying the term number by three and adding two gives five at term one, eight at term two, and so on. This lets you find a distant term without listing every earlier one.
| Term number | Value |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
Check the starting position
Check where numbering begins. If the first listed term is term one, substitute one when testing the start. Using zero instead would shift the whole pattern, even if the difference between consecutive values stayed correct.
Test a proposed rule against several supplied rows, then apply it in the stated operation order. Matching a few rows checks those rows; the given rule is what justifies using the same calculation at a later term.
See the method in action.
Worked example 1
A table has values 4, 7, 10, 13 for term numbers 1, 2, 3, 4. Use the rule “term number ×3, then add 1” to find term 20.
Worked solution
- 1
Check the supplied rule at term one.
The first output matches the first listed value.
- 2
Check another listed term.
The rule also agrees with the fourth row.
- 3
Use term number twenty.
The rule acts on the term number, so term 20 has value 61.
61.
Worked example 2
A number machine doubles the input and then subtracts 1. Find the outputs for inputs 1, 4 and 10.
Worked solution
- 1
Apply the machine to input one.
Double the input before subtracting one.
- 2
Apply it to input four.
Use the same operation order for every input.
- 3
Apply it to input ten.
The three outputs in input order are 1, 7 and 19.
1, 7 and 19.
Try it yourself.
Check your understanding
A rule is “multiply the term number by 5, then subtract 2”. Find term 6.
Show answer & reasoning
28.
Use the term number in the rule: . Multiply before subtracting.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- Use “, then add 1” to find term 5.1 mark
- A sequence starts at 3 and increases by 2. Find term 6.2 marks
- Check whether “, then add 3” produces 5,7,9 for terms 1,2,3.2 marks
- A rule is “, then add 0.5”. Find the output for input 8.1 mark
- A row of joined squares uses 4 sticks at stage 1, 7 at stage 2 and 10 at stage 3. Each extra square adds 3 sticks. How many sticks at stage 8?3 marks
- A table gives 6,10,14 for terms 1,2,3. Give a term-number rule and test it.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Use “, then add 1” to find term 5.1 mark
.
21. - A sequence starts at 3 and increases by 2. Find term 6.2 marks
The terms are 3,5,7,9,11,13; there are five increases.
13. - Check whether “, then add 3” produces 5,7,9 for terms 1,2,3.2 marks
; ; .
Yes. - A rule is “, then add 0.5”. Find the output for input 8.1 mark
.
4.5. - A row of joined squares uses 4 sticks at stage 1, 7 at stage 2 and 10 at stage 3. Each extra square adds 3 sticks. How many sticks at stage 8?3 marks
Start with 4 and add seven groups of 3: .
25 sticks. - A table gives 6,10,14 for terms 1,2,3. Give a term-number rule and test it.3 marks
; ; .
Multiply the term number by 4, then add 2.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Growing rational patterns; rules; brackets/order of operations; unknowns; algorithms.
Official curriculum source ↗