See the idea
Before you start
Recognise repeated changes and distinguish an item’s position from its value. Multiplication helps when describing a rule directly in terms of position.
Words used in this guide
- Sequence
- An ordered list of terms.
- Term-to-term rule
- A rule telling how to get from one term to the next.
- Position-to-term rule
- A rule giving a term directly from its position, often using n.
Why the method works
A repeated difference explains how a sequence grows, but the starting value matters too. For 6, 10, 14, 18, the difference is 4. Multiples of 4 are 4, 8, 12, 16; every sequence term is two more, so the nth term is 4n + 2 when positions begin at n = 1. A short list alone does not uniquely determine a rule, so state any assumed continuation.
Describe the change
In 4, 7, 10, 13, each term is three more than the previous one. Under that rule, the next term is 16. A short list can fit more than one possible rule, so state the rule you are using.
Separate position and value
In this sequence the first term is 4, not 3. Three times the position gives 3, 6, 9, 12; adding 1 gives 4, 7, 10, 13. So the nth term is 3n + 1 for positive whole-number positions.
Test the rule
Try n = 1 and n = 4: 3 × 1 + 1 = 4, and 3 × 4 + 1 = 13. Then find the tenth term without listing every term: 3 × 10 + 1 = 31.
Worked examples, step by step
Example 1: A sequence follows the rule 2n + 5. Find term 6.
- Substitute n = 6.
- 2 × 6 + 5 = 17.
Example 2: Find the tenth term of 6, 10, 14, 18 under the add-4 rule.
- The direct rule is 4n + 2. Substitute n = 10.
- 4 × 10 + 2 = 42.
- Check the rule on known terms: n = 1 gives 6 and n = 4 gives 18.
Example 3: A sequence follows 5n − 1. Is 34 a term?
- Set 5n − 1 = 34. Add 1 to get 5n = 35, then divide by 5.
- n = 7, a positive whole-number position, so 34 is the seventh term.
- If the calculation gave a non-whole-number position, 34 would not be a term under this rule.
Example 4: Tiles form rows of 3, 6, 12 and 24. Under a doubling rule, what comes next?
- Each term is multiplied by 2, not increased by one fixed difference.
- The next term is 48.
- The differences 3, 6 and 12 are changing. Do not force a constant-difference rule onto a multiplicative pattern.
Common mistakes and how to repair them
“The difference is 3, so the nth term is always 3n.”
That works for 3, 6, 9, … but not for 5, 8, 11, …, whose rule is 3n + 2. Check the first term.
“The next term is the only possible answer.”
A finite list can fit several rules. In a school task, use the stated or clearly intended rule and explain your assumption.
Check your understanding
Choose an answer for each question. You can check again after reading the explanation.
Worked answers
Under the add-3 rule, what follows 4, 7, 10, 13? 16. 13 + 3 = 16.
For the rule 3n + 1, what is term 5? 16. 3 × 5 + 1 = 16.
Practise independently
Work these out on paper. Include a diagram or a sentence where the question asks you to explain. Open each solution after making an attempt.
3. Under the add-5 rule, continue 2, 7, 12, 17.
Show the reasoning
22, then 27. Add five at each step.
4. Find term 8 when the rule is 3n + 2.
Show the reasoning
26, because 3 × 8 + 2 = 26.
5. Find a direct rule for 7, 11, 15, 19 under constant differences.
Show the reasoning
4n + 3 for positions starting at 1. Check n = 1: 4 + 3 = 7.
6. Under the rule 4n + 3, can 30 be a term?
Show the reasoning
No. 4n = 27 gives n = 6.75, not a positive whole-number position.
If a method still feels uncertain, revisit the closest worked example, then try a fresh problem. A correct answer copied from a solution tells you less than an explanation you can give independently.
Learn it → Try it → Explain it
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Questions students and parents ask
Does n always start at 1?
Not always. Many school sequences use n = 1 for the first term; some contexts start at 0. State the indexing convention.
Can a sequence decrease?
Yes. For example, 20, 17, 14, 11 decreases by 3 and has rule 23 − 3n when n starts at 1.
Why test more than one term?
A wrong rule may accidentally match one term. Test several given positions, including the first, before using it to predict a later value.
Help at home: Build and label a growing pattern
Build groups of 4, 7, 10 and 13 counters. Label positions 1, 2, 3 and 4 separately from the counter counts.
Ask what is added each time and what the first group contains. If the child writes 3n, compare its first output with the first group of four.
Ask for a far-away term and then reverse the question: “Could 25 belong?” Under 3n + 1, it is term 8.
Keep the conversation focused on the reasoning you can hear or see. One short activity cannot establish lasting mastery or diagnose a learning difficulty.
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