A pattern begins:
5, 8, 11, 14, …
What is the 100th term?
Listing ninety-six more terms would work in principle.
Algebra gives a better route.
An algebraic expression is a compressed rule that links a pattern position to the quantity generated at that position.
For this sequence:
the first term is 5 and the common difference is 3.
At position n:
5+3(n−1)=3n+2.
So the 100th term is:
3(100)+2=302.
The quick answer: separate the changing part from the fixed part
- Identify the position variable, usually n.
- Find how much the pattern changes when n increases by 1.
- Build the variable part from that repeated growth.
- Find the fixed adjustment needed to match the first few cases.
- Test the expression against several known terms or diagrams.
Position and value are different quantities
For 5, 8, 11, 14:
- position 1 has value 5;
- position 2 has value 8;
- position 3 has value 11;
- position 4 has value 14.
The variable n represents position, not the term value itself.
This distinction becomes essential when writing nth-term expressions.
Use a table to expose the relationship
For the same sequence:
- n=1 → 5;
- n=2 → 8;
- n=3 → 11;
- n=4 → 14.
Compare each value with 3n:
- 3(1)=3, need +2;
- 3(2)=6, need +2;
- 3(3)=9, need +2.
So the rule is 3n+2.
A table can reveal the algebra by separating the contribution from the position from the constant adjustment that remains.
Worked sequence example
7, 12, 17, 22, …
Common difference=5.
Start with 5n.
At n=1, 5n=5 but term is 7.
Need +2.
Rule:
5n+2.
Check n=4:
20+2=22.
Growing diagrams reveal why the coefficient appears
Suppose a row of connected squares grows by adding one square at each stage.
If the number of squares is n, then the shape contains n square units.
That is already the expression:
n.
But if each square contributes three new matchsticks because one side is shared with the previous square, then matchstick count may follow:
4, 7, 10, 13, …
Rule:
3n+1.
The +1 comes from the starting edge structure; the 3n comes from repeated growth.
A geometrical explanation is stronger than numerical fitting
If a learner merely notices 4,7,10,13 and writes 3n+1, the rule works.
If the learner can also explain:
“Each new square contributes three external matchsticks, plus one starting matchstick remains fixed,”
the expression is tied to the diagram’s construction.
Perimeter expressions from variable dimensions
A rectangle has width x cm and length x+5 cm.
Perimeter:
2x+2(x+5).
Expand and collect:
4x+10.
The expression tells us how perimeter changes as x changes.
Area expressions preserve multiplication
For the same rectangle:
area=x(x+5).
Expanded:
x²+5x.
Perimeter and area use the same dimensions but different relationships.
That is why diagram-to-algebra translation must identify what quantity is being modelled, not only the labels on the shape.
Worked diagram example: border tiles
Imagine an n-by-n square of interior tiles with a one-tile-wide border added around it.
Outer square side length:
n+2.
Outer area:
(n+2)².
Interior area:
n².
Border tiles:
(n+2)²−n².
Expand:
n²+4n+4−n².
Result:
4n+4.
The algebra reveals a linear border-count rule hidden inside two areas.
Different decompositions can give equivalent expressions
The same border could be counted as:
- top row: n+2;
- bottom row: n+2;
- left side excluding corners: n;
- right side excluding corners: n.
Total:
(n+2)+(n+2)+n+n.
=4n+4.
Two counting methods produce equivalent algebra.
When two correct decompositions describe the same diagram, their algebraic expressions should simplify to the same value.
Use small cases to test a formula
For border rule 4n+4:
If n=1:
4(1)+4=8.
A 1-by-1 centre inside a 3-by-3 outer square indeed has 8 border tiles.
If n=2:
12 border tiles.
Draw and count to confirm.
Reverse problem: find the stage from the expression
If a matchstick pattern has rule 3n+1, which stage uses 100 matchsticks?
3n+1=100.
3n=99.
n=33.
Generalisation becomes an equation when the value is known and the position is unknown.
Non-linear patterns need different structure
1,4,9,16,25,…
These are square numbers.
Rule:
n².
The first differences are not constant:
3,5,7,9,…
A linear form an+b would not fit all terms.
Pattern type should be diagnosed before choosing the algebraic family.
Triangular-number structure as an extension
Dot patterns may grow:
1,3,6,10,15,…
The nth triangular number is:
n(n+1)/2.
This is beyond simple constant-difference pattern work but shows how geometry and counting can lead naturally to quadratic expressions.
Do not force an nth-term rule from too little evidence
A finite set of values can fit more than one mathematical rule.
In school contexts, the intended rule is normally supported by:
- the construction of the diagram;
- a stated pattern;
- a simplest consistent relationship.
When a diagram is available, derive from its mechanism rather than fitting numbers blindly.
Common misconception 1: n is the term value
n usually represents position or stage number.
Common misconception 2: coefficient equals the first term
For arithmetic sequences, the coefficient of n is linked to common difference, not generally the first term.
Common misconception 3: every growing pattern is linear
Square and other patterns may require quadratic or more complex expressions.
Common misconception 4: visual counting can ignore shared edges or corners
Overcounting often occurs when pieces overlap or share boundaries.
Common misconception 5: one matching case proves the rule
Test several cases and connect the expression to the generating mechanism.
A pattern-to-algebra diagnostic ladder
- Can the learner separate position from value?
- Can the learner build a position-value table?
- Can the learner identify constant first differences?
- Can the learner form an+ b rules?
- Can the learner derive the constant adjustment?
- Can the learner read variable dimensions from a diagram?
- Can the learner form perimeter and area expressions?
- Can the learner account for shared edges and corners?
- Can the learner derive one expression in two different ways?
- Can the learner test the rule on small cases?
- Can the learner recognise when a linear rule is insufficient?
How this fits Secondary Mathematics
Forming expressions from patterns and diagrams connects sequences, algebraic language, geometry, functions, graphs and proof. It trains the learner to move from examples to a general relationship rather than remain dependent on case-by-case counting.
Exact G2 and G3 depth should be checked against the relevant current SEAB syllabus.
The deeper lesson: algebra preserves the growth mechanism
A pattern can be drawn, counted or tabulated.
An expression compresses all those cases into one rule.
The strongest algebraic expression does not merely fit the visible examples. It explains how the structure grows, which part changes with n, and which part remains fixed as every new case is generated.