Sequences and the nth term

The method

1. Find how much the sequence changes by each time, then look at the differences of those differences.
2. For a linear sequence the nth term has the form an + b, where a is the common difference.
3. Compare the sequence with an to find the constant b.
4. For a quadratic sequence compare with an², where a is half the second difference, then find the remaining linear part.
5. Substitute n = 1, 2 and 3 back into your formula to check it gives the first three terms.

Mistakes that lose marks

• Writing 3n as the whole answer when the sequence starts at 5, instead of comparing with 3n and adding the constant.
• Halving the second difference and then forgetting to compare with an² to find the rest of the formula.
• Assuming a sequence is linear without working out the second difference.
• Saying a number is not in the sequence when the equation gives a whole-number value of n.
• Giving the terms of the sequence instead of the nth term when the question asks for the formula.

7 exam-style questions

These are our own questions written in the style of the exam, not past-paper questions. The mark codes are M = method, A = accuracy and B = independent mark. Try each one, then open the solution.

1. Exam-style core Find the nth term of the sequence 5, 8, 11, 14, ... [3 marks]
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   1. The terms go up by 3 each time, so the nth term starts with 3n M1
   2. Compare with 3n: 3, 6, 9, 12, ... and each term is 2 more than this M1
   3. So the nth term is 3n + 2 A1

   Answer: 3n + 2

2. Exam-style core A sequence has nth term 4n − 7. (a) Find the 15th term. (b) Which term of the sequence is 41? [4 marks]
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   1. (a) Substitute n = 15: 4 × 15 − 7 M1
   2. (a) = 60 − 7 = 53 A1
   3. (b) Solve 4n − 7 = 41, so 4n = 48 M1
   4. (b) n = 12, so 41 is the 12th term A1

   Answer: (a) 53 (b) the 12th term

3. Exam-style core Is 100 a term of the sequence with nth term 3n + 4? Show your working. [3 marks]
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   1. Suppose 3n + 4 = 100 M1
   2. 3n = 96, so n = 32 M1
   3. 32 is a whole number, so yes: 100 is the 32nd term A1

   Answer: Yes, 100 is the 32nd term

4. Exam-style extended Find the nth term of the sequence 2, 7, 14, 23, 34, ... [4 marks]
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   1. First differences are 5, 7, 9, 11 and the second difference is 2, so the nth term contains n² M1
   2. Subtract n² = 1, 4, 9, 16, 25 from the terms: 1, 3, 5, 7, 9 M1
   3. The remainder is the linear sequence 2n − 1 A1
   4. So the nth term is n² + 2n − 1 A1

   Answer: n² + 2n − 1

5. Exam-style extended Find the nth term of the sequence 3, 10, 21, 36, 55, ... [4 marks]
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   1. First differences are 7, 11, 15, 19 and the second difference is 4, so the nth term contains 2n² M1
   2. Subtract 2n² = 2, 8, 18, 32, 50 from the terms: 1, 2, 3, 4, 5 M1
   3. The remainder is the linear sequence n A1
   4. So the nth term is 2n² + n A1

   Answer: 2n² + n

6. Exam-style extended The nth term of a sequence is n² + 3n. (a) Find the value of n for which the term is 40. (b) Decide whether 88 is a term of the sequence. [4 marks]
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   1. (a) n² + 3n = 40, so n² + 3n − 40 = 0 M1
   2. (a) (n + 8)(n − 5) = 0, so n = 5 A1
   3. (b) n² + 3n = 88, so n² + 3n − 88 = 0 and (n + 11)(n − 8) = 0 M1
   4. (b) n = 8, so 88 is the 8th term A1

   Answer: (a) n = 5 (b) yes, 88 is the 8th term

7. Exam-style higher A gardener plants trees in rows. The first row has 3 trees, the second row has 8, the third row has 15 and the fourth row has 24. (a) Find the nth term for the number of trees in row n. (b) How many trees are in row 12? (c) Which row has 143 trees? [5 marks]
   Show full solution

   1. First differences are 5, 7, 9 and the second difference is 2, so the nth term contains n² M1
   2. Subtract n² = 1, 4, 9, 16 from the terms: 2, 4, 6, 8, so the remainder is 2n M1
   3. (a) The nth term is n² + 2n A1
   4. (b) Row 12: 12² + 2 × 12 = 144 + 24 = 168 A1
   5. (c) n² + 2n = 143, so (n + 1)² = 144 and n = 11 (reject n = −13 because a row number must be positive) A1

   Answer: (a) n² + 2n (b) 168 trees (c) row 11

Where this is in the syllabus

Sequences and the nth term appears in Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1. For the official specifications and past papers, use the board's own website: Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1.

Questions students ask