A23: Generating Sequences
Generate terms of a sequence from term-to-term or position-to-term rules
📋 Key Concepts
Sequence: An ordered list of numbers. Each number is called a term.
Two types of rules:
- Term-to-term: How to get from one term to the next
- Position-to-term: Formula for the nth term
📝 Term-to-Term Rules
Definition: A rule that tells you how to find the next term from the previous term.
Example 1
Generate the first 5 terms of a sequence starting with 3, where each term is found by adding 5.
Solution:
Start: 3
Add 5: 3 + 5 = 8
Add 5: 8 + 5 = 13
Add 5: 13 + 5 = 18
Add 5: 18 + 5 = 23
Sequence: 3, 8, 13, 18, 23, ...
Example 2
Generate the first 6 terms starting with 2, where each term is found by multiplying by 3.
Solution:
2, 6, 18, 54, 162, 486, ...
Example 3
A sequence starts 1, 4, 10, 22, ... Find the term-to-term rule.
Solution:
4 - 1 = 3
10 - 4 = 6
22 - 10 = 12
Differences: 3, 6, 12 (doubling each time)
Rule: Multiply by 2 and add 2
Check: 4 × 2 + 2 = 10 ✓, 10 × 2 + 2 = 22 ✓
📝 Position-to-Term Rules
Definition: A rule that gives the nth term directly. Written as a formula using n.
Notation:
- n = position in sequence (1st, 2nd, 3rd, ...)
- uₙ or aₙ = nth term
Example 4
Find the first 5 terms of the sequence where uₙ = 2n + 1
Solution:
u₁ = 2(1) + 1 = 3
u₂ = 2(2) + 1 = 5
u₃ = 2(3) + 1 = 7
u₄ = 2(4) + 1 = 9
u₅ = 2(5) + 1 = 11
Sequence: 3, 5, 7, 9, 11, ...
Example 5
Find the 10th term of the sequence uₙ = n² - 3
Solution:
u₁₀ = 10² - 3 = 100 - 3 = 97
📝 Finding a Specific Term
Example 6
The nth term of a sequence is 3n - 2. Find the 20th term.
Solution:
u₂₀ = 3(20) - 2 = 60 - 2 = 58
Example 7
Is 100 a term in the sequence uₙ = 4n + 7?
Solution:
Set 4n + 7 = 100
4n = 93
n = 23.25
n is not a whole number, so 100 is NOT in the sequence.
📝 Finding the Position of a Term
Example 8
The nth term is 5n - 3. Which term equals 47?
Solution:
5n - 3 = 47
5n = 50
n = 10
47 is the 10th term
🧮 Worked Examples — Exam Style
Sequences come in two rule types: term-to-term (use the previous term) and position-to-term (the nth term — substitute n directly).
A sequence starts at 3 with term-to-term rule 'add 7'. Write the first four terms.
3, 10, 17, 24 — keep adding 7.
B1 for all four. Term-to-term = repeated application; nth term = formula.
For uₙ = 5n − 2, find the first three terms and the 50th term.
n = 1: 3; n = 2: 8; n = 3: 13. 50th: 5 × 50 − 2 = 248
M1 for substituting, A1 for 248. The nth term makes the 50th term as easy as the 1st — that is its whole point.
Exam wording to expect: "Generate the first … terms", "Find the …th term", "Is … a term in the sequence?".
❓ Practice Questions
Q1: Generate the first 5 terms starting at 2, adding 4 each time.
Q2: Generate the first 4 terms starting at 1, multiplying by 2 each time.
Q3: Find the first 5 terms where uₙ = 3n - 1.
Q4: Find the 15th term of uₙ = 2n + 5.
Q5: Is 50 a term in the sequence uₙ = 6n + 2?
Q6: Which term in the sequence uₙ = 7n - 4 equals 66?
Q7: Find the first 5 terms of uₙ = 4n − 3
Q8: Which term of uₙ = 9n − 7 equals 101?
✅ Answers
- 2, 6, 10, 14, 18
- 1, 2, 4, 8
- 2, 5, 8, 11, 14
- 35
- 6n + 2 = 50 → n = 8, so yes, 50 is the 8th term
- n = 10 (the 10th term)
🧮 Worked Examples — Harder Exam Style
A sequence has uₙ = n² − 3. Find the first four terms and state whether 46 is a term.
n = 1: 1 − 3 = −2; n = 2: 4 − 3 = 1; n = 3: 9 − 3 = 6; n = 4: 16 − 3 = 13
Terms: −2, 1, 6, 13
n² − 3 = 46 → n² = 49 → n = 7 (positive root) → yes, 46 is the 7th term
M1 for substituting, A1 for the terms, A1 for n = 7. Quadratic nth terms appear at Foundation as pattern tables — square numbers minus/plus a constant.
📋 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. 2, 6, 10, 14, 18 (add 4 each time)
Q2. 1, 2, 4, 8 (double each time)
Q3. n = 1…5: 2, 5, 8, 11, 14
Q4. 2(15) + 5 = 35
Q5. 6n + 2 = 50 → 6n = 48 → n = 8, so yes — the 8th term
Q6. 7n − 4 = 66 → 7n = 70 → n = 10
Q7. 1, 5, 9, 13, 17
Q8. 9n − 7 = 101 → 9n = 108 → n = 12
📌 Key Facts to Memorise
- Term-to-term: state the RULE (add 4, double, subtract 3)
- nth term: substitute n = 1, 2, 3… directly
- "Is k a term?" → solve k = un for n; only WHOLE positive n counts
- Linear sequences change by a constant amount; geometric multiply by a constant
- uₙ notation: u₅ means the 5th term — substitute n = 5
🗺️ In the Exam
Generating sequences: 2–3 marks on Paper 1. The "is k a term?" version adds an AO2 layer (solve and interpret a non-integer n as "no").
Links to: nth term (A25), types of sequences (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
- Term-to-term: how to get from one term to the next
- Position-to-term: formula using n for the nth term
- n must be a whole number (positive integer)
- To check if a number is in the sequence, solve for n
- If n isn't a whole number, the term isn't in the sequence
🧠 Problem-Solving Strategies
Problem-Solving
For term-to-term rules: look at differences or ratios between consecutive terms. For position-to-term rules: substitute the position number into the formula. To check if a number belongs, set the formula equal to it and solve — n must be a positive integer.
Multi-Step Problem
The nth term of a sequence is 4n - 3. (a) Find the first 4 terms. (b) Is 97 in the sequence? (c) Which term equals 97?
Solution:
(a) n=1: 1, n=2: 5, n=3: 9, n=4: 13
(b) Set 4n - 3 = 97 → 4n = 100 → n = 25. Yes, it's the 25th term.
(c) 97 is the 25th term
⚠️ Common Errors
Watch Out!
1. Wrong: Thinking the first term corresponds to n = 0 Correct: The first term always corresponds to n = 1
2. Wrong: Saying 50 is in the sequence 6n + 2 because 50 "looks right" Correct: Check algebraically: 6n + 2 = 50 → n = 8 (whole number), so yes it is the 8th term
3. Wrong: Confusing term-to-term with position-to-term rules Correct: Term-to-term tells how to get the next term; position-to-term gives a formula for any term directly
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A sequence has nth term 5n + 3. (a) Find the 1st, 10th and 100th terms. (b) Is 2023 in the sequence? Show working. (c) Two consecutive terms add to 73. Which two terms are they?
(a) 1st: 5(1) + 3 = 8. 10th: 5(10) + 3 = 53. 100th: 5(100) + 3 = 503
(b) 5n + 3 = 2023 → 5n = 2020 → n = 404. Yes, it's the 404th term.
(c) Consecutive terms: (5n + 3) + (5(n+1) + 3) = 73 → 10n + 11 = 73 → 10n = 62 → n = 6.2. Not a whole number, so no two consecutive terms add to 73. Wait — let me recheck: 10n + 8 + 5 + 3 = 10n + 11. Hmm: (5n+3) + (5n+8) = 10n + 11 = 73 → n = 6.2. No integer solution. Actually this means 73 is not achievable.
Mark scheme: (a) 2 marks. (b) 2 marks with working. (c) 2 marks for correct method and conclusion.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A savings account adds £15 per month. In month 1 the balance is £35.
(a) Write a formula for the balance B in month n.
(b) When does the balance exceed £500?
(c) Explain why this model might not be realistic long-term.
Answers: (a) B = 35 + 15(n - 1) = 15n + 20. (b) 15n + 20 > 500 → n > 32. So from month 33. (c) The model ignores interest earned and inflation; in reality savings grow with compound interest, not just linear deposits.
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