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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Substitute n = 1 into the formula: a 1 = 4(1) - 3 = 1
- S2
Substitute n = 2 into the formula: a 2 = 4(2) - 3 = 5
- S3
Substitute n = 3 into the formula: a 3 = 4(3) - 3 = 9
- S4
Substitute n = 4 into the formula: a 4 = 4(4) - 3 = 13
- S5
Substitute n = 5 into the formula: a 5 = 4(5) - 3 = 17
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Substitute n = 1 into the formula: a 1 = 4(1) - 3 = 1
- S2
Substitute n = 2 into the formula: a 2 = 4(2) - 3 = 5
- S3
Substitute n = 3 into the formula: a 3 = 4(3) - 3 = 9
- S4
Substitute n = 4 into the formula: a 4 = 4(4) - 3 = 13
- S5
Substitute n = 5 into the formula: a 5 = 4(5) - 3 = 17
Final answer: The first five terms are 1, 5, 9, 13, 17.
Common mistakes
- Confusing the n th term formula with the recursive relation. — 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. — 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. — 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. — 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. — 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. — Take care with algebraic manipulations. Simplify expressions step-by-step to avoid mistakes.
Where the marks go
- Full worked solution (all marking points)5 marks