A sequence begins:
7, 11, 15, 19, 23, …
What is the 100th term?
Continuing the pattern ninety-five more times is possible.
It is not efficient.
A sequence becomes algebraically powerful when we stop asking only “What comes next?” and start asking “What term appears at position n?”
The first step is to notice that the sequence increases by 4 each time.
That is a term-to-term rule:
add 4.
But a term-to-term rule still requires us to know the previous term.
For a distant term, we need a position-to-term rule.
The quick answer: build the rule from position
For the sequence 7,11,15,19,…:
- common difference = 4;
- start with 4n;
- at n=1, 4n=4 but the first term is 7;
- we need +3.
So the nth term is:
4n+3.
Now the 100th term is:
4(100)+3=403.
Term-to-term and position-to-term rules do different jobs
For 7,11,15,19,…:
term-to-term rule:
add 4.
Position-to-term rule:
4n+3.
The first tells us how to move locally.
The second tells us where any position sits globally.
A recursive rule describes movement from one term to the next. An nth-term rule describes the whole sequence from the viewpoint of position.
Arithmetic sequences have constant first difference
A sequence is arithmetic when consecutive terms differ by a constant amount.
Example:
3,8,13,18,23,…
Differences:
5,5,5,5,…
The common difference is 5.
General arithmetic nth term
If the first term is a and common difference is d, then:
Tₙ=a+(n−1)d.
Why?
From the first term to the nth term, we make n−1 equal jumps.
Each jump contributes d.
So:
first term + number of jumps × jump size.
Worked example using a+(n−1)d
Sequence:
12,17,22,27,…
a=12.
d=5.
Tₙ=12+5(n−1).
=12+5n−5.
=5n+7.
Both forms are equivalent.
A table makes the linear rule visible
For 12,17,22,27:
- n=1 → 12;
- n=2 → 17;
- n=3 → 22;
- n=4 → 27.
Compare with 5n:
- 5→need +7;
- 10→need +7;
- 15→need +7.
So 5n+7.
The table separates the variable growth from the fixed offset.
Negative common difference
Sequence:
30,24,18,12,6,…
Common difference:
−6.
Use:
Tₙ=30+(n−1)(−6).
=30−6n+6.
=36−6n.
Check n=1:
36−6=30.
Fractional sequences
Sequence:
1/2, 3/4, 1, 5/4, …
Common difference:
1/4.
Tₙ=1/2+(n−1)/4.
=(2+n−1)/4.
=(n+1)/4.
Sequences are not restricted to whole numbers.
Reverse question: is a value in the sequence?
Sequence nth term:
4n+3.
Is 155 a term?
Solve:
4n+3=155.
4n=152.
n=38.
Because n is a positive integer, 155 is the 38th term.
A non-integer position means the value is not a term
Is 154 in 4n+3?
4n+3=154.
n=151/4=37.75.
Sequence positions are positive integers.
So 154 is not a term of this sequence.
Reverse sequence questions are equations with an extra condition: the position must belong to the allowed index set.
Two sequences can intersect
Suppose:
Aₙ=3n+1
and:
Bₙ=5n−7.
At the same position, when are they equal?
3n+1=5n−7.
8=2n.
n=4.
Both equal 13 at position 4.
This connects sequence rules to simultaneous or intersection thinking.
Sequence graphs
An arithmetic sequence with Tₙ=4n+3 corresponds to points:
- (1,7);
- (2,11);
- (3,15);
- (4,19).
These points lie on the straight line y=4x+3.
But the sequence usually uses only integer position values, so the graph is discrete points rather than the entire continuous line.
Not every sequence is arithmetic
Consider:
2,4,8,16,32,…
Differences:
2,4,8,16,…
Not constant.
The terms double each time.
This is geometric, not arithmetic.
A linear nth-term form an+b will not fit it.
Square-number sequence
1,4,9,16,25,…
Rule:
n².
The first differences are 3,5,7,9,…, not constant.
This signals a non-linear sequence family.
Second differences as an extension
For a quadratic sequence, first differences change but second differences are constant.
Example:
2,6,12,20,30,…
First differences:
4,6,8,10.
Second differences:
2,2,2.
Indeed the sequence is n²+n.
This is a useful extension into higher algebraic pattern recognition.
Do not infer too much from too few terms
Any finite list of terms can be fitted by many mathematical rules if enough complexity is allowed.
School sequence problems usually signal an intended simple structure through:
- constant differences;
- constant ratios;
- diagram growth;
- known number families;
- a stated recurrence.
Use the simplest rule supported by the construction and context.
Checking an nth-term rule
A proposed rule should reproduce multiple known terms.
For 4n+3:
- n=1 →7;
- n=2 →11;
- n=5 →23.
Testing only the first term is weak because many wrong rules can match one point.
Common misconception 1: nth term means next term
The nth term gives a rule for an arbitrary position n, not only the next term.
Common misconception 2: common difference is the constant term
For arithmetic sequences, common difference becomes the coefficient of n after simplification, not generally the constant offset.
Common misconception 3: use the first term as the coefficient of n
The coefficient is determined by the rate of change across positions.
Common misconception 4: every sequence is linear
Geometric, quadratic and other sequence families need different rules.
Common misconception 5: any real n is a valid sequence position
Ordinary sequence positions are positive integers unless a different domain is explicitly defined.
A sequence diagnostic ladder
- Can the learner extend a pattern correctly?
- Can the learner identify first differences?
- Can the learner distinguish term-to-term from position-to-term rules?
- Can the learner form an arithmetic nth term?
- Can the learner use a+(n−1)d?
- Can the learner simplify the rule?
- Can the learner find a distant term directly?
- Can the learner solve reverse position questions?
- Can the learner enforce integer position conditions?
- Can the learner connect an arithmetic sequence to discrete points on a line?
- Can the learner recognise when the sequence is not arithmetic?
How this fits Secondary Mathematics
Sequences connect number patterns, algebraic expressions, functions, graphs and later series. The central shift is from local continuation to generalisation: instead of generating one more term, the learner describes the rule for every allowed position.
The deeper lesson: general rules compress time
A term-to-term rule walks through the sequence.
An nth-term rule jumps directly to the destination.
The mathematical power of the nth term is that one expression replaces an indefinitely long list: position enters, the rule acts, and the corresponding term appears without rebuilding every step that came before it.