A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Sequences and Series: nth Term and Recursive Relations — mark scheme explained
The short answer
Sequences are a fundamental part of A-Level Mathematics, particularly in the study of sequences and series. This topic covers various types of sequences, including those given by a formula for the n th term and those generated by a simple relation of the form x n+1 = f(x n ).
The question
Find the first five terms of the sequence defined by a n = 4n - 3.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving sequences, marks are typically awarded for correct substitution into formulas, accurate calculation of terms, and correct identification of sequence properties (increasing, decreasing, periodic).
What the command words demand
- Find
- Calculate the required terms of a sequence using the given formula or relation.
- Determine
- Identify whether a sequence is increasing, decreasing, or periodic based on its terms.
- List
- Write down the first few terms of a sequence to observe patterns.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 2-3 minutes per mark. For a 5-mark question, spend approximately 10-15 minutes.
- Substitute n = 1 into the formula: a 1 = 4(1) - 3 = 11 mark
- Substitute n = 2 into the formula: a 2 = 4(2) - 3 = 51 mark
- Substitute n = 3 into the formula: a 3 = 4(3) - 3 = 91 mark
- Substitute n = 4 into the formula: a 4 = 4(4) - 3 = 131 mark
- Substitute n = 5 into the formula: a 5 = 4(5) - 3 = 171 mark
Final answer: The first five terms are 1, 5, 9, 13, 17.
Work through every step correctly and you earn all 5 marks.
Another worked example
Find the first four terms of the sequence defined by x n+1 = x n + 2, with x 1 = 3.
- Substitute n = 1 into the recursive relation: x 2 = x 1 + 2 = 3 + 2 = 52 marks
- Substitute n = 2 into the recursive relation: x 3 = x 2 + 2 = 5 + 2 = 71 mark
- Substitute n = 3 into the recursive relation: x 4 = x 3 + 2 = 7 + 2 = 91 mark
Final answer: The first four terms are 3, 5, 7, 9.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Confusing the n th term formula with the recursive relation.
Why it happens: Students sometimes mix up the explicit formula for the n th term and the recursive relation, leading to incorrect calculations.
Fix: Always identify whether the sequence is defined explicitly or recursively. Use the appropriate method to find terms.
Forgetting to substitute the initial value in a recursive relation.
Why it happens: Students often forget to use the given initial value when working with recursive relations, leading to incorrect sequences.
Fix: Always start by substituting the initial value into the recursive relation and proceed step-by-step.
Incorrectly identifying whether a sequence is increasing or decreasing.
Why it happens: Students may compare terms incorrectly, leading to wrong conclusions about the nature of the sequence.
Fix: Calculate and list the first few terms of the sequence. Compare each term with the previous one to determine if it is increasing or decreasing.
Failing to recognize periodic sequences.
Why it happens: Students may not notice patterns in sequences, leading to incorrect identification of periodicity.
Fix: Calculate and list the first few terms. Look for repeating patterns to identify if the sequence is periodic.
Using the wrong formula or relation when finding terms.
Why it happens: Students may use an incorrect formula or relation, leading to incorrect results.
Fix: Double-check the given formula or relation. Ensure you are using the correct method for the type of sequence (explicit or recursive).
Not simplifying expressions correctly when finding terms.
Why it happens: Students may make algebraic errors when substituting values into formulas, leading to incorrect results.
Fix: Take care with algebraic manipulations. Simplify expressions step-by-step to avoid mistakes.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Generate Sequence Terms | Substitute n = 1 to 5 into the nth term formula to find each term. | 5 |
| Recursive Sequence Terms | Generate the first four terms by repeatedly applying the recurrence relation from the given start. | 4 |
| Increasing or Decreasing Sequence | Compare consecutive terms to decide whether the sequence increases or decreases. | 3 |
| Classify Sequence Behaviour | Decide whether the given sequence is increasing or decreasing by comparing successive terms. | 3 |
| Identify Periodic Sequence | Test whether the sequence repeats and, if it does, state its period. | 4 |
| Total across these question types | 19 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.