SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 17
A pattern becomes mathematics when we can state what changes, what stays fixed and how any position can be found. Listing the next term is useful, but generalisation goes further: it replaces repeated counting with a rule.
Secondary Mathematics increasingly asks learners to move from examples to structure. A sequence can be represented as a list, a table, a diagram or an algebraic rule. The most powerful representation depends on the question.
This guide develops arithmetic sequences, nth-term rules, constant differences, decreasing and fractional sequences, shape patterns, reverse-position questions and the difference between recursive and direct rules. It is an independent learning companion; exact pacing varies across subject levels and schools. Return to the Secondary Mathematics Hub.
Useful prior guides: Algebraic Expressions and Variables, Algebraic Formulae, Substitution and Rearrangement, and Coordinates, Linear Graphs and Relationships.
1. A sequence has terms in positions
A sequence is an ordered list. The first term is at position 1, the second at position 2, and so on.
For 5, 8, 11, 14, … the first term is 5 and the common difference is 3.
Position is not the same as value
At position 4, the value is 14. Confusing “fourth term” with “term equal to 4” is an early source of error.
2. Constant first difference signals an arithmetic sequence
An arithmetic sequence changes by the same amount between consecutive terms.
Example: 12, 17, 22, 27, … has common difference +5.
Example: 30, 24, 18, 12, … has common difference −6.
Check several gaps
Do not decide from only the first two terms. Confirm that the same difference continues across the displayed sequence.
3. The nth term gives the value at any position
For an arithmetic sequence with first term a and common difference d, the nth term is:
a + (n − 1)d.
This formula says: start at the first term and make n − 1 equal jumps of size d.
Worked example
For 5, 8, 11, 14, …, a = 5 and d = 3.
nth term = 5 + 3(n − 1) = 5 + 3n − 3 = 3n + 2.
4. Another route: multiply by the difference, then adjust
For 5, 8, 11, 14, … the difference is 3, so begin with 3n.
| n | 3n | actual term |
|---|---|---|
| 1 | 3 | 5 |
| 2 | 6 | 8 |
| 3 | 9 | 11 |
The actual term is always 2 larger, so the rule is 3n + 2.
This route and a + (n − 1)d are equivalent.
5. Verify a rule against more than one term
A proposed rule n + 4 gives 5 at n = 1, so it matches the first term of 5, 8, 11, … but fails at n = 2 because it gives 6 instead of 8.
A correct rule must reproduce every term in the sequence, not just one chosen example.
Fast verification
Test n = 1, a middle displayed position and one later position. This catches many incorrect rules quickly.
6. Decreasing arithmetic sequences work the same way
Consider 41, 35, 29, 23, …
a = 41 and d = −6.
nth term = 41 − 6(n − 1) = 41 − 6n + 6 = 47 − 6n.
Check
At n = 4, 47 − 24 = 23, matching the sequence.
7. Fractional and decimal differences are still valid
Consider 1.5, 2.0, 2.5, 3.0, … The common difference is 0.5.
nth term = 1.5 + 0.5(n − 1) = 0.5n + 1.
Arithmetic sequences are about constant change, not about whole numbers only.
8. Find a term directly without listing every term
For the sequence with nth term 7n − 3, the 50th term is:
7(50) − 3 = 350 − 3 = 347.
The direct rule removes the need to generate the preceding 49 terms.
9. Reverse questions ask for the position
Does 101 occur in the sequence with nth term 4n + 1?
Set 4n + 1 = 101. Then 4n = 100 and n = 25. Since 25 is a positive whole-number position, 101 is the 25th term.
Non-example
Does 100 occur? 4n + 1 = 100 gives n = 24.75. Since a standard sequence position is a positive integer, 100 is not a term.
10. Recursive and direct rules answer different jobs
A recursive rule tells how to get the next term from the current one: “start at 5 and add 3 each time”.
A direct rule tells the value at position n: “3n + 2”.
Recursive rules are useful for generating the next few terms. Direct rules are efficient for distant positions and reverse-position problems.
11. Tables connect position to value
| Position n | Term T |
|---|---|
| 1 | 4 |
| 2 | 9 |
| 3 | 14 |
| 4 | 19 |
The output increases by 5 when position increases by 1. The rule is T = 5n − 1.
This is also a linear relationship between n and T. Plotting the points produces a straight-line pattern when the positions are viewed numerically.
12. Shape patterns can be translated into algebra
Suppose Figure 1 uses 4 tiles, Figure 2 uses 7, Figure 3 uses 10 and Figure 4 uses 13.
The number of tiles increases by 3 each time, so T = 3n + 1.
Structure explanation
A stronger solution explains why: perhaps every figure contains three tiles for each stage plus one fixed corner tile. The algebra then reflects visible structure rather than only numerical fitting.
13. Different visual decompositions can give equivalent rules
A shape pattern may be counted as 2n + n + 4, or as 3n + 4. These are equivalent because 2n + n simplifies to 3n.
Two students can therefore use different decompositions and still describe the same invariant structure.
Useful comparison
Ask whether the formulas agree algebraically and reproduce the same figures.
14. Constant first difference is not universal
The sequence 1, 4, 9, 16, 25, … has first differences 3, 5, 7, 9. They are not constant.
This is the square-number sequence n², not an arithmetic sequence.
Boundary
Do not force every pattern into an + b form. Linear nth-term methods apply when the relationship is linear.
15. Differences can help classify a pattern
For 2, 6, 12, 20, 30, … the first differences are 4, 6, 8, 10. The second differences are constant at 2.
This suggests a quadratic pattern; indeed the terms follow n(n + 1).
Treat systematic second-difference methods as an extension if they have not yet appeared in the learner’s course. The main habit is to avoid assuming linearity too quickly.
16. Missing-term problems can use local or global structure
Sequence: 14, 19, __, 29, 34.
The constant difference is 5, so the missing term is 24.
Alternatively, the nth-term rule is 5n + 9. At n = 3 it gives 24.
17. A sequence rule is a model with a domain
If n represents figure number, n normally takes positive whole-number values 1, 2, 3, …
The expression 3n + 2 can be evaluated at n = 2.5 algebraically, but Figure 2.5 may not exist in the original pattern.
This is a useful bridge to domain in functions and graphs.
18. Common pattern errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Finds only the next term | Recursive thinking only | How can you find the 100th term directly? |
| Uses difference as nth term | Adjustment missing | What does dn give at n = 1? |
| Accepts non-integer position | Sequence domain ignored | Can a term occur at position 8.4? |
| Assumes every sequence is linear | Pattern family not checked | Are the first differences constant? |
| Formula matches one term only | Verification too weak | Does it reproduce several displayed terms? |
19. Practice laboratory
- Find the common difference of 7, 12, 17, 22, …
- Write the next three terms of 30, 26, 22, 18, …
- Find the nth term of 4, 7, 10, 13, …
- Find the nth term of 20, 15, 10, 5, …
- Find the 40th term of 6n − 1.
- Does 145 occur in the sequence 6n + 1?
- Does 100 occur in the sequence 6n + 1?
- Find the nth term of 2.5, 3.0, 3.5, 4.0, …
- A tile pattern uses 5, 9, 13, 17 tiles. Find a rule for Figure n.
- Find the missing term: 11, 18, __, 32, 39.
- State whether 1,4,9,16,… is arithmetic.
- For T = 8n − 3, find n when T = 77.
- Explain the difference between “add 4 each time” and “4n + 1”.
- A sequence has first term 13 and common difference 7. Find its nth term.
- Find the first term of the sequence with nth term 9n − 5.
- Find the common difference of the sequence with nth term 9n − 5.
20. Explained answers
1. 5.
2. 14, 10, 6.
3. 3n + 1.
4. 25 − 5n.
5. 6(40) − 1 = 239.
6. 6n + 1 = 145 gives 6n = 144, n = 24, so yes.
7. 6n + 1 = 100 gives n = 16.5, so no standard sequence position.
8. 0.5n + 2.
9. 4n + 1.
10. 25.
11. No; differences are 3,5,7,…
12. 8n − 3 = 77 gives 8n = 80, so n = 10.
13. The first is recursive; the second gives the term directly from its position.
14. 13 + 7(n−1) = 7n + 6.
15. n = 1 gives 4.
16. 9.
21. Complete mixed problem
A theatre seating model has 18 seats in Row 1, 22 in Row 2, 26 in Row 3 and continues with the same increase.
The common difference is 4, so Row n has 18 + 4(n−1) = 4n + 14 seats.
Row 25 has 4(25) + 14 = 114 seats.
Which row has 150 seats? Solve 4n + 14 = 150. Then 4n = 136 and n = 34.
The same rule answers a forward question and a reverse question.
22. Teaching patterns as structure, not guessing
Ask the learner to describe the repeated change, then to connect position and value in a two-column table. Only after that should the algebraic rule be compressed.
For shape patterns, ask what part grows with n and what part stays fixed. This often produces a formula naturally.
Changed-case test
If one extra fixed tile is added to every figure, what changes in the nth-term rule? The coefficient of n may remain the same while the constant changes. This reveals which part represents growth and which represents a fixed offset.
23. Questions students often ask
Is the nth term always linear?
No. Only linear sequences have a rule of the form an + b.
Why do we use n − 1 in a + (n − 1)d?
Because the first term requires zero jumps from the starting value, the second one jump, and position n requires n − 1 jumps.
Can a sequence have negative terms?
Yes. A decreasing arithmetic sequence may cross zero and continue into negative values.
What is the best rule check?
Substitute several positions, especially n = 1 and a later term.
24. Return path and sources
Sequences connect arithmetic change to algebraic generalisation. Revisit Algebraic Expressions and Variables if simplifying rules is weak and Coordinates, Linear Graphs and Relationships to see linear sequences as input-output relationships.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact pattern and sequence scope varies across subject levels and schools. All examples here are independently written teaching examples.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Observe the change, identify the invariant structure, generalise by position, test the rule and return it to the original pattern.
