SPEC A23, A25 (F) • MODULE 2: ALGEBRA • ⭐ HIGH PRIORITY
nth Term of Linear Sequences
Spot the constant difference, write the rule, jump to any term.
BOTH TIERS • ALL PAPERS • LESSON 2.26
What Is a Linear Sequence?
A linear (or arithmetic) sequence goes up — or down — by the same amount each step. That step size is the common difference, written d.
...
Here d = 4 and the first term is 3.
Test for linear
Work out the differences between consecutive terms. If they're all the same, the sequence is linear. If the differences grow or shrink, it isn't linear (it might be quadratic — covered in lesson 2.27).
d can be negative
"19, 15, 11, 7, …" has d = −4. The formula works the same — just keep track of the sign.
Two Kinds of Rule — Know the Difference
A sequence can be described in two ways. Both are valid — but they do different jobs. The exam will usually ask for the nth term.
Term-to-Term Rule
Describes how to get the next term from the current one.
"Start at 3, add 4 each time"
Pro: easy to state.
Con: to find the 100th term you'd have to grind out 99 steps.
Position-to-Term (nth term)
A formula for the n-th term directly, given its position.
nth term = 4n − 1
Pro: jump straight to term 100 (or 1000) — just substitute.
Con: needs a bit of algebra to write down.
Exam cue
"Find the nth term of the sequence…" → you need the position-to-term rule. "Find the next three terms" → use term-to-term (or substitute n = 5, 6, 7 into the nth term). Read the command word carefully.
The nth Term Formula
Every linear sequence has an nth term of the form dn + c. The two bits you need are the common difference d and the constant c.
The formula
nth term = dn + (first term − d)So the constant c = (first term − d), not the first term itself. This is a very common error on nth-term questions — even after finding d correctly.
Why this works
At n = 1 the formula gives d(1) + (first − d) = first term. ✓ At n = 2 it gives d(2) + (first − d) = first + d. ✓ Each time n increases by 1, the value goes up by d — exactly the common difference.
Equivalent form
You may also see nth term = a + (n − 1)d, where a is the first term. It's the same formula rearranged — use whichever you remember.
Walkthrough: 5, 9, 13, 17, ...
Apply the formula step by step.
...
STEP 1 Find d. Differences: 9 − 5, 13 − 9, 17 − 13 — all equal 4. d = 4.
STEP 2 Find c. c = first term − d = 5 − 4 = 1.
STEP 3 Write the rule: nth term = 4n + 1.
Verify (always do this)
n = 1: 4(1) + 1 = 5 ✓ | n = 2: 4(2) + 1 = 9 ✓ | n = 3: 4(3) + 1 = 13 ✓
Always Verify at n = 1
Before writing your final answer, substitute n = 1 into your nth term rule. If it doesn't give the first term of the sequence, your rule is wrong.
The 10-second self-check
You wrote: nth term = 4n + 1. Substitute n = 1: 4(1) + 1 = 5. First term of the sequence is 5. Match ✓ — rule is correct.
Example of a wrong rule
Suppose you accidentally wrote "nth term = 4n + 5" (using the first term as the constant). Check: 4(1) + 5 = 9. But the first term is 5, not 9. Rule is wrong — go back and fix c.
Why this matters
In the exam, a correct rule is usually worth 2 marks. A wrong rule may still score a method mark if d was identified correctly, but you'll lose the accuracy mark — and a 10-second n = 1 check catches almost every slip.
Jumping to Any Term
Once you have the nth term rule, finding the 50th or 100th term is just a substitution — no more listing.
Example: The nth term of a sequence is 4n + 1. Find the 50th term.
Substitute n = 50
50th term = 4(50) + 1 = 200 + 1 = 201
The advantage of the nth term rule
With a term-to-term rule ("start at 5, add 4"), finding the 50th term means 49 additions. With the nth term rule, it's one substitution. That's why the nth term is the exam's preferred form.
MCE Worked Example01
Find the nth term of the sequence 4, 9, 14, 19, 24, …. Then use your rule to find the 20th term. [3 marks]
M
Method: find the common difference d, then c = first term − d. Write nth term as dn + c. Substitute n = 20 for the 20th term.
C
Differences: 9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5. So d = 5.
C
c = first term − d = 4 − 5 = −1.
C
nth term = 5n − 1. Check n = 1: 5(1) − 1 = 4 ✓
C
20th term: 5(20) − 1 = 100 − 1 = 99.
E
nth term = 5n − 1; 20th term = 99.
Reverse: "Is X in the Sequence?"
Given the nth term rule, you can work out which position (if any) a given value occupies. Set the rule equal to the value and solve for n.
Example: The nth term of a sequence is 4n + 1. Is 61 a term in this sequence?
Method
Set 4n + 1 = 61 → 4n = 60 → n = 15.
n = 15 is a positive integer → yes, 61 is the 15th term.
The rule for "yes, it's in"
After solving for n, check: is n a positive integer (1, 2, 3, …)? If yes, the value is in the sequence at that position. If n is a fraction or zero or negative, the value is not in the sequence.
"n = 15" ≠ "15th term"
Students sometimes answer "n = 15" as if that's the answer. But 15 is the position, not the term value. If the question asks "Is 61 in the sequence? Justify", answer "Yes — 61 is the 15th term (n = 15 is a positive integer)."
Examiner Alert: The Constant c
The biggest trap on nth term questions: using the first term as the constant instead of (first term − d).
Weak Student
Sequence: 4, 7, 10, 13, …
d = 3, first term = 4. Writes:
"nth term = 3n + 4"
Check n = 1: 3(1) + 4 = 7. But the first term is 4, not 7. Wrong rule.
Strong Student
Sequence: 4, 7, 10, 13, …
d = 3, c = first − d = 4 − 3 = 1. Writes:
"nth term = 3n + 1"
Check n = 1: 3(1) + 1 = 4 ✓. Correct.
Examiner alert
The constant in the nth term formula is (first term − d), not just the first term. This is a very frequent error on nth-term questions. The fix is trivial: always verify by substituting n = 1 — the formula should give back the first term.
Examiner Alert: Position vs Term Value
The second common trap: confusing the position n with the value at that position.
Weak Student
Q: "Find the 50th term of the sequence with nth term 5n + 2."
Writes:
"n = 50"
50 is the position, not the term. Has to substitute to find the actual value.
Strong Student
Q: "Find the 50th term of the sequence with nth term 5n + 2."
Substitutes n = 50:
"50th term = 5(50) + 2 = 252"
Reports the value. Full marks.
Examiner alert
The question "Find the 50th term" wants a number. Giving n = 50 is telling the examiner which term you're looking for — you still need to compute its value. Also: a question like "Which term is 252?" wants the position n = 50, not the term value.
Examiner Alert: Start at n = 1
Standard convention
The first term corresponds to n = 1, not n = 0. Always. This is the GCSE convention and it matches the formula nth term = dn + c when c = first − d.
Weak
"Check first term of my rule 3n + 1 by substituting n = 0:"
3(0) + 1 = 1
Thinks the first term should be 1. But if the sequence starts at 4, they now think the rule is wrong — when actually it's right.
Strong
"Check first term by substituting n = 1:"
3(1) + 1 = 4 ✓
Matches the sequence's actual first term. Rule verified.
In short
n = 1 gives the 1st term. n = 2 gives the 2nd. n = 100 gives the 100th. There is no n = 0 term in the standard convention.
Sequences from Patterns
The spec notes that nth-term questions can be set from patterns and diagrams. Just count the objects in each pattern to get the sequence — then proceed as normal.
Convert pattern → sequence → nth term
Pattern counts: 3, 5, 7, 9, 11, … | d = 2 | c = first − d = 3 − 2 = 1
nth term = 2n + 1
Check n = 1: 2(1) + 1 = 3 ✓ | How many dots in Pattern 20? 2(20) + 1 = 41 dots.
The workflow
Count objects in each pattern → write the numbers as a sequence → find d → find c → write nth term → verify. Same routine as any linear-sequence question.
Your Turn
Here is a linear sequence: 8, 15, 22, 29, 36, …
(a) Find an expression for the nth term of the sequence. [2 marks]
(b) Is 150 a term in this sequence? Justify your answer. [2 marks]
M
(a) Find d, then c = first − d, then nth term = dn + c. Verify at n = 1.
C
Differences: 15 − 8 = 7, 22 − 15 = 7, 29 − 22 = 7. d = 7. | c = 8 − 7 = 1.
C
nth term = 7n + 1. Check n = 1: 7(1) + 1 = 8 ✓
E
(a) nth term = 7n + 1
M
(b) Set 7n + 1 = 150 and solve for n. Check whether n is a positive integer.
C
7n + 1 = 150 → 7n = 149 → n = 149/7 ≈ 21.29
E
(b) No, 150 is not in the sequence because n = 149/7 is not a positive integer. (The 21st term is 148 and the 22nd is 155 — 150 falls between.)
Common Errors To Avoid
Students often writeCorrect versionWhy it matters
For 4, 7, 10, … : nth term = 3n + 4nth term = 3n + 1 (c = first − d = 4 − 3)Using first term as constant — most common slip
Verify with n = 0Verify with n = 1 — n starts at 1Standard GCSE convention
"Find the 50th term" → answer: n = 5050th term = d(50) + c (substitute and compute)n is the position, not the value
"Which term is 252?" → answer: 252Solve for n → answer is the position, e.g. n = 50Reverse confusion — position vs value
"Is 150 in the sequence?" → yes because it's big enoughSolve nth term = 150 for n; check n is a positive integerJustification needs the algebra
Only the term-to-term rule given when nth term askednth term must be dn + c — an expression in nWrong form of rule
Quick Self-Check
Before moving on: Does my rule pass the n = 1 check? Am I reporting the term value when asked for a term, or the position n when asked which term? For "is X in the sequence", did I solve for n and check it's a positive integer?
Summary
The nth Term in Three Steps
Find d — the common difference between consecutive terms (it must be constant for a linear sequence).
Find c — the constant: c = first term − d.
Write the rule : nth term = dn + c. Verify by substituting n = 1 — it must give the first term.
Three Types of Question
Generate terms from a rule — substitute n = 1, 2, 3, … into the nth term (or apply the term-to-term rule repeatedly).
Find the nth term of a given sequence — spot d and c; verify at n = 1.
Is X in the sequence / which term is X? — solve "nth term = X" for n; answer is in the sequence only if n is a positive integer.
Spec Coverage
Spec refContent
A23Generate terms of a sequence from either a term-to-term or a position-to-term rule (including from patterns and diagrams).
A25 (F)Deduce expressions to calculate the nth term of linear sequences.
Higher tier extends A25 to quadratic sequences — covered in lesson 2.27.
Exam Tip
This is a ⭐ high priority topic — every paper, both tiers. The nth term question is usually worth 2 marks; a follow-up ("find the 50th term" or "is X in the sequence") adds 1–2 more. A clean three-step method plus the n = 1 verification check should earn full marks every time.
ExamRizz GCSE Maths (AQA 8300) • Lesson 2.26 • Module 2: Algebra • Spec A23, A25 (F) • Foundation + Higher • ⭐ High priority
Next: What Is a Linear Sequence?