Pattern spotting is useful, but Mathematics asks for something stronger than “I can see what comes next”. A general term describes the value of any position in a sequence without generating every earlier term first.
This guide develops the move from visual or numerical pattern to algebraic rule. It shows how position matters, how first differences reveal structure, why checking only one or two terms is weak evidence, and how a general term can be tested, interpreted and used backwards.
The key shift: from next term to nth term
Consider the sequence 5, 8, 11, 14, 17, … Most students can predict 20 next. That uses local pattern recognition: add 3.
The general-term question is different: what is the value of the 100th term?
If the first term is 5 and every step adds 3, then the nth term is:
5 + 3(n-1)
= 3n + 2Now the 100th term is 302 without writing the first 99 terms.
Why position must appear in the rule
A sequence is not only a list of values. Each value is attached to a position:
| Position n | Term |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
The rule 3n+2 maps position to value. That is why sequence work is also early function thinking.
Worked example 1: linear sequence
Find the nth term of 7, 12, 17, 22, …
The common difference is 5, so begin with 5n. This gives 5, 10, 15, 20, … Compare with the target sequence: every term is 2 larger.
Tn = 5n + 2Check: n=1 gives 7; n=4 gives 22. The rule survives more than one position.
Worked example 2: decreasing sequence
Find the nth term of 20, 16, 12, 8, …
The difference is -4. Start with -4n: -4, -8, -12, -16. To reach 20, 16, 12, 8, add 24.
Tn = 24 - 4nThe negative coefficient records the decreasing pattern.
Worked example 3: use the rule backwards
For Tn=4n+1, which position contains 101?
4n+1=101
4n=100
n=25So 101 is the 25th term. General terms do not only generate values; they let us solve membership questions.
Visual patterns need a counted structure
Suppose Figure 1 has 4 tiles, Figure 2 has 7, Figure 3 has 10 and Figure 4 has 13. The visible growth is +3 each stage. A useful explanation is not merely “add three more tiles”. It is “there are three tiles for each stage number, plus one fixed tile”, giving 3n+1.
This is stronger because the algebra is tied to the construction of the figure.
A general term should explain more than the next term
Many rules can fit a few starting terms. A strong rule should match the intended structure of the sequence and survive additional tests.
Checking positions 1, 2 and 10 is better than checking only the first term. For a visual sequence, connect the algebra to parts of the diagram as well.
Difference tables as a structural clue
Constant first differences indicate a linear sequence. Constant second differences suggest a quadratic sequence. Difference tables do not prove the intended model by themselves, but they are a strong diagnostic tool.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Can find next term but not nth term | Local pattern only | Link each term to its position n |
| Uses difference as full rule | Missing offset | Compare dn with actual first term |
| Checks only n=1 | Weak verification | Test several separated positions |
| Cannot explain visual rule | Algebra detached from structure | Count growing and fixed parts separately |
Tutor route: from observation to generalisation
- Generate the next two terms.
- Record position and term in a table.
- Inspect first differences.
- Build a candidate rule from the difference and offset.
- Check several positions.
- Use the rule to find a distant term.
- Reverse the rule to test whether a value belongs to the sequence.
Practice
- Find the nth term of 4, 9, 14, 19, …
- Find the nth term of 30, 27, 24, 21, …
- For Tn=6n-1, find the 40th term.
- For Tn=7n+3, determine whether 150 is in the sequence.
Answers
1. 5n-1. 2. 33-3n. 3. 239. 4. 7n+3=150 gives n=147/7=21, so yes, 150 is the 21st term.
Connected routes
Continue to Arithmetic Sequences: Constant Difference as Structure, Quadratic Patterns: Detecting Constant Second Differences, and Recursive Rules and Explicit Rules. Return to the Mathematics Learning Hub.