A25: nth Term
Find the nth term of linear sequences; quadratic sequences (Higher)
📋 Key Concepts
nth term: A formula that allows you to find any term in a sequence without working out all the previous terms.
Position numbers: The position of each term (1st = position 1, 2nd = position 2, etc.)
📝 Finding nth Term of Linear Sequences
Method:
- Find the common difference (d)
- Multiply position (n) by the difference: dn
- Find what to add/subtract by comparing to dn
- Write as: dn + c
Example 1
Find the nth term of: 5, 8, 11, 14, 17, ...
Solution:
Step 1: Find the difference: 8 - 5 = 3
Step 2: Start with 3n: 3, 6, 9, 12, 15
Step 3: Compare to sequence: 5, 8, 11, 14, 17
Difference: +2, +2, +2, +2, +2
nth term: 3n + 2
Example 2
Find the nth term of: 20, 17, 14, 11, 8, ...
Solution:
Difference: -3
Start with -3n: -3, -6, -9, -12, -15
Compare: 20, 17, 14, 11, 8
Difference: +23 (since 20 = -3 + 23)
nth term: -3n + 23 or 23 - 3n
Example 3
Find the nth term of: 3, 7, 11, 15, 19, ...
Solution:
Difference: 4
4n: 4, 8, 12, 16, 20
Compare: 3, 7, 11, 15, 19
Difference: -1
nth term: 4n - 1
📝 Using the nth Term
Example 4
Find the 50th term of the sequence with nth term 4n - 1.
Solution:
Substitute n = 50:
4(50) - 1 = 200 - 1 = 199
Example 5
Is 100 a term in the sequence 3n + 5?
Solution:
Set 3n + 5 = 100
3n = 95
n = 31.67 (not a whole number)
100 is NOT in the sequence
📝 Finding nth Term of Quadratic Sequences (Higher)
Method:
- Find first differences
- Find second differences (constant for quadratic)
- Half of second difference = coefficient of n²
- Subtract n² term from sequence to find remainder
Example 6
Find the nth term of: 4, 7, 12, 19, 28, ...
Solution:
Sequence: 4, 7, 12, 19, 28
1st diff: 3, 5, 7, 9
2nd diff: 2, 2, 2 (constant)
Half of 2nd diff = 1, so start with n²
n²: 1, 4, 9, 16, 25
Sequence - n²: 3, 3, 3, 3, 3
nth term: n² + 3
Example 7
Find the nth term of: 5, 9, 15, 23, 33, ...
Solution:
Sequence: 5, 9, 15, 23, 33
1st diff: 4, 6, 8, 10
2nd diff: 2, 2, 2
Half = 1, so n² term
n²: 1, 4, 9, 16, 25
Sequence - n²: 4, 5, 6, 7, 8 (linear: n + 3)
nth term: n² + n + 3
Example 8
Find the nth term of: 3, 9, 19, 33, 51, ...
Solution:
1st diff: 6, 10, 14, 18
2nd diff: 4, 4, 4
Half = 2, so 2n²
2n²: 2, 8, 18, 32, 50
Sequence - 2n²: 1, 1, 1, 1, 1
nth term: 2n² + 1
📝 More Complex Quadratic Sequences
Example 9
Find the nth term of: 6, 15, 28, 45, 66, ...
Solution:
1st diff: 9, 13, 17, 21
2nd diff: 4, 4, 4
Half = 2, so 2n²
2n²: 2, 8, 18, 32, 50
Sequence - 2n²: 4, 7, 10, 13, 16 (linear: 3n + 1)
nth term: 2n² + 3n + 1
🧮 Worked Examples — Exam Style
Linear nth term: coefficient = common difference; constant = zero-term adjustment. Quadratic: second difference ÷ 2 gives the n² coefficient.
Find the nth term of 7, 10, 13, 16, …
Difference 3 → 3n; compare with 3n (3, 6, 9, 12): need +4 → 3n + 4
M1 for 3n, A1 for 3n + 4. The compare-with-3n method avoids sign slips on descending sequences.
Find the nth term of 11, 8, 5, 2, …
Difference −3 → −3n; compare with −3n (−3, −6, −9…): need +14 → −3n + 14
M1 for −3n, A1 for −3n + 14. Descending sequences: negative coefficient, positive constant.
Exam wording to expect: "find an expression for the nth term", "use your nth term to find…", quadratic versions at Higher.
❓ Practice Questions
Q1: Find the nth term of: 6, 10, 14, 18, 22, ...
Q2: Find the nth term of: 50, 46, 42, 38, 34, ...
Q3: Find the 100th term of: 7n - 3
Q4: Is 80 a term in the sequence 5n + 10?
Q5: Find the nth term of: 2, 5, 10, 17, 26, ...
Q6: Find the nth term of: 6, 11, 18, 27, 38, ...
Q7: Find the nth term of 9, 13, 17, 21, … and the 40th term
Q8: Which term of 6n − 3 is the first above 100?
✅ Answers
- 4n + 2
- -4n + 54 or 54 - 4n
- 697
- 5n + 10 = 80 → n = 14, yes it's the 14th term
- n² + 1
- n² + 2n + 3
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Difference 4 → 4n + 2 (check: n=1 → 6? No: 4 + 2 = 6 ≠ 6? terms are 6,10,14 — given sequence starts 6: 4n + 2 ✓)
Q2. Difference −4 → −4n + 54
Q3. 7 × 100 − 3 = 697
Q4. 5n + 10 = 80 → n = 14 → yes, the 14th term
Q5. Differences 3, 5, 7 → second difference 2 → n² + 1
Q6. Differences 5, 7, 9 → second difference 2 → n² + 2n + 3?? check n=1: 6 ✓ (1+2+3=6) → n² + 2n + 3
Q7. Difference 4 → 4n + 5; 40th = 165
Q8. 6n − 3 > 100 → n > 17.2 → 18th term
📌 Key Facts to Memorise
- Linear: an + b — a = common difference, b = term₀ (first term − a)
- Compare method: write an, line it under the sequence, the gap is b
- Quadratic: coefficient of n² = half the second difference; finish by matching
- "Is k a term?" — solve an + b = k; whole-number n required
- Check with n = 1 AND n = 2 before trusting the formula
🗺️ In the Exam
Nth term: 3–5 marks — finding it, using it, solving back. Quadratic nth terms are the Higher-boundary discriminator in this folder.
Links to: generating sequences (A23), types (A24), solving equations (A17).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Linear: Find difference, then multiply n by this and adjust
- Quadratic: Check second difference are constant
- Half of second difference = coefficient of n²
- Always check by substituting n = 1, 2, 3
- For linear, quick check: 1st term shows if constant is right
🧠 Problem-Solving Strategies
Problem-Solving
For linear sequences: find the difference (d), start with dn, then compare to find the constant. For quadratic: find second difference, halve it for the n² coefficient, then subtract n² terms to find the linear remainder. Always check with n = 1, 2, 3.
Multi-Step Problem
A pattern of dots forms: 4, 9, 16, 25, 36, ... (a) Find the nth term. (b) How many dots in the 50th pattern? (c) Is 400 in the sequence?
Solution:
(a) 1st diff: 5, 7, 9, 11. 2nd diff: 2, 2, 2. Half = 1, so n² term. n²: 1, 4, 9, 16, 25. Actual - n²: 3, 5, 7, 9, 11. This is 2n + 1. So nth term = n² + 2n + 1 = (n + 1)²
(b) (50 + 1)² = 51² = 2601
(c) 400 = (n + 1)² → n + 1 = 20 → n = 19. Yes, 400 is the 19th term.
⚠️ Common Errors
Watch Out!
1. Wrong: For a decreasing sequence like 20, 17, 14, ... the nth term is 3n + 17 Correct: Difference is -3, so nth term is -3n + 23 (check: n=1 gives 20 ✓)
2. Wrong: For quadratic sequences, using the first difference as the coefficient of n² Correct: Use HALF of the SECOND difference as the coefficient of n²
3. Wrong: Saying 100 is in the sequence 3n - 1 because it "looks about right" Correct: Check: 3n - 1 = 100 → n = 33.67 (not integer). 100 is NOT in the sequence.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A pattern uses tiles: 7, 11, 17, 25, 35, ... (a) Show this is a quadratic sequence. (b) Find the nth term. (c) How many tiles are needed for the 20th pattern? (d) Is 200 tiles possible?
(a) 1st diff: 4, 6, 8, 10. 2nd diff: 2, 2, 2. Constant second difference = quadratic.
(b) Half of 2nd diff = 1, so n². n²: 1, 4, 9, 16, 25. Remainder: 6, 7, 8, 9, 10. This is n + 5. So nth term = n² + n + 5
(c) 20² + 20 + 5 = 400 + 20 + 5 = 425 tiles
(d) n² + n + 5 = 200 → n² + n - 195 = 0 → n = (-1 + √(1+780))/2 = (-1 + √781)/2 ≈ (-1 + 27.95)/2 ≈ 13.47. Not a whole number, so 200 is not possible.
Mark scheme: (a) 1 mark. (b) 2 marks. (c) 1 mark. (d) 2 marks for solving and conclusion.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The number of seats in each row of a theatre follows a pattern: 20, 24, 30, 38, 48, ...
(a) Find the nth term.
(b) There are 25 rows. How many seats in the last row?
(c) The fire limit is 60 seats per row. Are any rows over the limit?
Answers: (a) 1st diff: 4, 6, 8, 10. 2nd diff: 2. Half = 1, so n². n²: 1, 4, 9, 16, 25. Remainder: 19, 20, 21, 22, 23 = n + 18. nth term = n² + n + 18. (b) Row 25: 625 + 25 + 18 = 668 seats. (c) Yes — from row 7 onwards (7² + 7 + 18 = 74 > 60). The theatre would need safety adjustments.
📝 Exam Questions by Topic
🎬 Video Resources
Ready to ace your GCSE Mathematics exams?
Get the best revision books and guides to boost your grades.