The idea, step by step
Two different kinds of rule
A term-to-term rule says how to move from one term to the next. “Add 3” explains 5, 8, 11, 14, but it does not state the first term. To generate the whole sequence you need both the starting value and the change.
A term-number rule starts with the position in the list. “Multiply the term number by 3, then add 2” gives the same sequence because term 1 is 3×1+2=5.
| Term number | Value |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
Check the starting position
Term number 1 means the first listed term. It is not automatically stage 0. A rule that works for term 2 but fails for term 1 needs correcting.
To test a proposed rule, use several rows. A practical rule must also match the situation: a tile pattern must give whole tiles, while a measured length might have a decimal value.
A common trap
The rule “add 4” always means term 10 is 14.
Instead: That is a step rule, not a rule based on the term number. You also need the first value; term 10 is reached after nine steps from term 1.
See the thinking
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.
- Check term 1: 1×3+1=4, which matches.
- For term 20, calculate 20×3+1.
Worked example 2
A number machine doubles the input and then subtracts 1. Find the outputs for inputs 1, 4 and 10.
- Input 1 gives 2×1−1=1.
- Input 4 gives 2×4−1=7; input 10 gives 2×10−1=19.
Check your understanding
A rule is “multiply the term number by 5, then subtract 2”. Find term 6.
Show answer
28.
6×5−2=30−2=28.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Use “term number ×4, 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 “term number ×2, then add 3” produces 5,7,9 for terms 1,2,3.2 marks
- A rule is “input ÷2, 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 “term number ×4, then add 1” to find term 5.1 mark
5×4+1=21.
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 “term number ×2, then add 3” produces 5,7,9 for terms 1,2,3.2 marks
1×2+3=5; 2×2+3=7; 3×2+3=9.
Yes. - A rule is “input ÷2, then add 0.5”. Find the output for input 8.1 mark
8÷2+0.5=4.5.
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: 4+21=25.
25 sticks. - A table gives 6,10,14 for terms 1,2,3. Give a term-number rule and test it.3 marks
1×4+2=6; 2×4+2=10; 3×4+2=14.
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 ↗