A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Arithmetic Sequences and Series — mark scheme explained
The short answer
Understanding and working with arithmetic sequences and series is a fundamental part of A-Level Mathematics. An arithmetic sequence (or arithmetic progression) is a sequence in which each term after the first is obtained by adding a constant, called the common difference ( d ), to the preceding term.
The question
Find the 12th term of the arithmetic sequence where the first term is 8 and the common difference is 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 arithmetic sequences and series, marks are typically awarded for correctly identifying the first term and common difference, using the correct formulas, and performing accurate calculations. Partial credit may be given for showing correct working even if the final answer is incorrect.
What the command words demand
- Find
- Calculate or determine a specific value or result.
- Calculate
- Perform mathematical operations to find a numerical answer.
- Determine
- Identify or establish a particular value, formula, or relationship.
- Show that
- Prove or demonstrate a given statement using appropriate steps and calculations.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 2-3 minutes per mark for questions on arithmetic sequences and series. This allows time to carefully identify the required values, apply the appropriate formula, and check your work.
- Identify the given values: a 1 = 8 , d = 3 , n = 120 marks
- Use the formula for the n-th term of an arithmetic sequence: a n = a 1 + (n - 1)d1 mark
- Substitute the values into the formula: a 12 = 8 + (12 - 1) × 31 mark
- Simplify the expression: a 12 = 8 + 11 × 3 = 8 + 33 = 411 mark
Final answer: The 12th term is 41.
Work through every step correctly and you earn all 3 marks.
Another worked example
Find the sum of the first 15 terms of an arithmetic series where the first term is 6 and the common difference is 4.
- Identify the given values: a 1 = 6 , d = 4 , n = 150 marks
- Use the formula for the sum of the first n terms of an arithmetic series: S n = n/2 [2a 1 + (n - 1)d]1 mark
- Substitute the values into the formula: S 15 = 15/2 [2 × 6 + (15 - 1) × 4]1 mark
- Simplify the expression inside the brackets: S 15 = 15/2 [12 + 56] = 15/2 × 681 mark
- Calculate the final sum: S 15 = 15 × 34 = 5101 mark
Final answer: The sum of the first 15 terms is 510.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Using the wrong formula for the n-th term or sum of an arithmetic series.
Why it happens: Students sometimes confuse the formulas for the n-th term and the sum of the first n terms. It's important to identify which formula is needed based on the given information.
Fix: Always double-check the problem statement to determine whether you need the n-th term or the sum of the series, and use the appropriate formula.
Forgetting to subtract 1 from n in the n-th term formula.
Why it happens: The formula for the n-th term is a n = a 1 + (n - 1)d . Students sometimes forget to subtract 1 from n, leading to incorrect results.
Fix: Always remember to use (n - 1) in the formula for the n-th term.
Incorrectly substituting values into the formulas.
Why it happens: Students may substitute the wrong values or make arithmetic errors when plugging numbers into the formulas.
Fix: Double-check your substitutions and calculations to ensure accuracy. Use a calculator if necessary.
Using the common difference instead of the first term in the sum formula.
Why it happens: The sum formula S n = n/2 [2a 1 + (n - 1)d] requires the first term and the common difference. Students sometimes use the common difference in place of the first term.
Fix: Ensure you are using the correct values for a 1 and d in the sum formula.
Forgetting to divide by 2 in the sum formula.
Why it happens: The sum formula involves dividing by 2, which students sometimes overlook, leading to incorrect results.
Fix: Always remember to divide by 2 when using the sum formula S n = n/2 [2a 1 + (n - 1)d] or S n = n/2 (a 1 + a n ) .
Using the wrong value for n in the formulas.
Why it happens: Students sometimes use the wrong value for n , especially when dealing with large numbers or multiple steps.
Fix: Always verify that you are using the correct value for n based on the problem statement.
Forgetting to simplify expressions before calculating the final answer.
Why it happens: Students may rush through calculations and forget to simplify expressions, leading to incorrect results.
Fix: Take your time to simplify expressions step-by-step before calculating the final answer.
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 |
|---|---|---|
| Nth Term Of Sequence | Find a specific term of an arithmetic sequence using the nth term formula. | 3 |
| Arithmetic Series Sum | Find the sum of the first 15 terms of a given arithmetic series. | 4 |
| Arithmetic Sequence Term | Use the nth term formula to find the common difference from two known terms. | 3 |
| Arithmetic Series Sum | Find the sum of a given number of terms using the first and last terms. | 3 |
| Total across these question types | 13 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.