A pattern is not understood when the next term is guessed correctly once. Mathematical generalisation asks what rule generates every term, how the rule can be expressed efficiently, and how we know the rule continues beyond the examples shown.
This Secondary 2 Mathematics Learning Guide develops numerical sequences, term-to-term change, position-to-term rules, first and second differences, simple figurate growth and nth-term generalisation. The emphasis is on moving from examples to a rule that can predict distant terms and be checked against the original sequence.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 6, Guide 2. Companion guides cover formulae and substitution, reasoning and proof, and coordinate geometry and locus.
Course boundary. Pattern and sequence work develops at different depths across lower secondary. This guide prioritises linear nth-term rules and uses constant second differences as an optional bridge into quadratic-pattern awareness, not as a demand that every learner has reached formal quadratic-sequence methods.
Navigate: sequence meaning · term-to-term rules · nth-term rules · difference structures · visual patterns · verification · practice and answers · teaching and transfer.
1. A sequence is an ordered list, not just a set of numbers
The sequence 4, 7, 10, 13, … has order. The first term is 4, the second 7, the third 10. Writing the same numbers in a different order creates a different sequence.
The position number is often written n. The term at that position may be written Tn. An nth-term rule connects position directly to term value.
Worked example 1: position matters
For 5, 9, 13, 17, …, the first term is 5 and the common difference is 4. The tenth term is not found most efficiently by adding 4 nine times by hand; a position rule gives direct access.
Because each step adds 4, term n is 5 + 4(n − 1), which simplifies to 4n + 1. At n = 10, the term is 41.
2. Term-to-term rules describe movement; nth-term rules describe position
For 3, 8, 13, 18, …, the term-to-term rule is add 5. This tells us how to move from one term to the next. The nth-term rule Tn = 5n − 2 tells us the value at any chosen position directly.
Both rules are useful, but they answer different questions. To find the next term, term-to-term is enough. To find the 100th term, the nth-term rule is much more efficient.
Worked example 2: derive from the first term and common difference
Sequence: 11, 16, 21, 26, … Common difference = 5. Starting from the first term, Tn = 11 + 5(n − 1) = 5n + 6.
Check: n = 1 gives 11; n = 2 gives 16; n = 4 gives 26. A proposed rule should reproduce several early terms before being trusted.
Worked example 3: decreasing sequence
Sequence: 20, 17, 14, 11, … Common difference = −3. Tn = 20 − 3(n − 1) = 23 − 3n.
A negative common difference is still a constant difference. The sequence is linear in position even though its values decrease.
3. Build a linear nth-term rule from gradient and intercept structure
A linear sequence with constant difference d has a rule of the form dn + c. The coefficient of n matches the common difference. The constant c is then chosen so the rule produces the first term.
Worked example 4: use the common difference as coefficient
Sequence: 7, 12, 17, 22, … Difference 5 suggests 5n + c. At n = 1, the term is 7, so 5 + c = 7 and c = 2. Hence Tn = 5n + 2.
This route connects sequences to linear graphs: position n acts like an input, term value acts like an output, and the common difference acts like gradient.
Worked example 5: identify whether a value belongs to the sequence
For Tn = 4n + 3, ask whether 79 is a term. Solve 4n + 3 = 79. Then 4n = 76 and n = 19. Since 19 is a positive whole-number position, 79 is the 19th term.
If solving gave n = 19.5, 79 would not occupy a term position in this sequence. The position domain matters.
Worked example 6: inverse use of an nth-term rule
For Tn = 6n − 1, which term equals 101? Solve 6n − 1 = 101. Then 6n = 102 and n = 17. The rule can predict terms from positions and recover positions from term values.
4. Difference tables help classify the pattern
If first differences are constant, a linear nth-term rule is appropriate. If first differences are not constant but second differences are constant, the pattern may be quadratic in n. Constant second differences are a signal, not a complete proof of a guessed formula.
Worked example 7: constant first differences
Sequence 2, 7, 12, 17 has first differences 5, 5, 5. It is linear, with Tn = 5n − 3.
Worked example 8: optional quadratic-pattern bridge
Sequence 3, 8, 15, 24, 35 has first differences 5, 7, 9, 11 and second differences 2, 2, 2. The terms fit Tn = n² + 2n.
Check n = 1 gives 3, n = 2 gives 8 and n = 5 gives 35. This section is a bridge: the important idea is recognising that non-constant first differences need a different family of rules.
A pattern can be non-polynomial
Sequence 3, 6, 12, 24, … multiplies by 2 each time. Its ratios are constant, not its differences. A linear nth-term rule would be unsuitable. The structural question is what remains constant: additive change, multiplicative change, alternating behaviour or something else?
5. Visual patterns should be converted into countable structure
Suppose Figure 1 uses 4 tiles, Figure 2 uses 7 and Figure 3 uses 10. The picture may change shape while the count increases by 3 each time. A useful decomposition is 3n + 1: three new tiles for each stage plus one persistent tile.
A visual rule is stronger when its components correspond to identifiable parts of the diagram rather than being fitted only to the first few totals.
Worked example 9: match formula components to a diagram story
A row pattern has n groups of 4 counters plus 2 fixed end counters. Total counters = 4n + 2. For n = 6, there are 26 counters.
The 4n and +2 terms have different geometric meanings. This makes the formula easier to explain and generalise.
Worked example 10: two valid-looking rules from too little data
Suppose only the first three terms 2, 4, 6 are shown. The obvious continuation is 8, 10, … with rule 2n. But another rule can be constructed that matches the first three terms and changes later.
Therefore finite data do not force one unique rule unless the pattern family or generating process is specified. In school sequences, the intended structure is usually constrained by context, differences or a diagram. This observation becomes important in the companion reasoning and proof guide.
6. Verify a general rule against several terms and one distant term
A proposed nth-term rule should reproduce the first term, at least one middle term and a later term. If it came from a visual pattern, explain why its components represent the construction process.
Worked example 11: reject a nearly correct rule
Sequence 6, 10, 14, 18. Someone proposes 4n + 6. At n = 1 this gives 10, not 6. The rule is immediately rejected. Correct rule: 4n + 2.
Checking the first term is often the fastest way to detect an intercept error.
Worked example 12: predict a distant term
For Tn = 7n − 4, the 50th term is 350 − 4 = 346. A term-to-term method would require 49 additions; the formula compresses the pattern.
7. Common errors reveal which part of generalisation is weak
- Next term correct but nth-term rule absent: term-to-term thinking only.
- Common difference used as entire rule: missing intercept/first-term adjustment.
- n begins at zero accidentally: repair position convention.
- Rule tested at only one term: weak verification.
- Fractional n accepted as a sequence position: repair domain awareness.
- Non-linear pattern forced into dn + c: repair classification.
- Visual formula fitted without explaining components: weak structural generalisation.
- Three examples treated as proof of one unique infinite pattern: repair reasoning boundary.
8. Practice: move from local change to a general rule
Questions 1–6. 1. Find the next two terms: 8, 13, 18, 23. 2. State the common difference. 3. Find the nth-term rule. 4. Find the 30th term. 5. Is 203 a term? 6. If so, state its position.
Questions 7–12. 7. Find the nth term of 20, 16, 12, 8, … 8. Find the 25th term. 9. Determine whether −80 is a term. 10. Sequence 4, 9, 16, 25, 36 has first differences 5, 7, 9, 11. State the second differences. 11. Give a formula that fits these terms. 12. Explain why a linear rule is unsuitable.
Questions 13–18. 13. A visual pattern has n groups of 5 objects plus 3 fixed objects. Write a formula. 14. Find the total at n = 12. 15. For Tn = 3n + 8, solve Tn = 68. 16. Is 69 a term? 17. A rule 6n + 1 is proposed for sequence 7, 13, 19, 25. Verify it. 18. Explain why observing 2, 4, 6 alone does not logically force every later term.
Explained answers 1–6
1. 28, 33. 2. 5. 3. Tn = 8 + 5(n − 1) = 5n + 3. 4. 153. 5. Solve 5n + 3 = 203, giving n = 40. 6. Yes, 40th term.
Explained answers 7–12
7. Difference −4, so Tn = 24 − 4n. 8. 24 − 100 = −76. 9. 24 − 4n = −80 gives n = 26, so yes. 10. Second differences are 2, 2, 2.
11. Tn = (n + 1)² fits: 4, 9, 16, 25, 36. 12. A linear sequence would have constant first differences; these first differences change.
Explained answers 13–18
13. 5n + 3. 14. 63. 15. 3n + 8 = 68 gives n = 20. 16. 3n + 8 = 69 gives n = 61/3, not a whole-number position, so 69 is not a term.
17. n = 1 gives 7, n = 2 gives 13, n = 4 gives 25, so the rule agrees with the shown linear sequence. 18. A finite initial list can be matched by more than one possible generating rule unless additional structure is specified.
9. Teaching sequence: notice → describe → express → verify → transfer
Begin with a sequence and ask what changes from term to term. Then ask how term value depends on position. Move from verbal description to a formula, and require the formula to be checked at several positions.
Next reverse the direction: provide a term value and ask whether it belongs to the sequence. Finally change the surface from a number list to a visual pattern or simple context while preserving the same algebraic structure.
Questions parents and tutors can ask
What is changing each time? Is the change additive or multiplicative? What does n represent? Does your rule work at n = 1? Can you use it for the 50th term? If I give you a term value, can you decide whether it actually belongs to the sequence?
10. The transfer test: from list to function
The sequence 6, 10, 14, 18, … has rule Tn = 4n + 2. Viewed as ordered pairs (1,6), (2,10), (3,14), these points lie on the straight line y = 4x + 2 when position is treated as x.
This connects sequence difference to graph gradient. The representation changes from list to equation to graph, but the same linear relationship survives.
Observe the change. Connect term value to position. Generalise with n. Verify several terms. Respect the position domain. Change representation without changing the relationship.
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