A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Geometric Sequences and Series — mark scheme explained
The short answer
Geometric sequences and series are fundamental concepts in mathematics, particularly useful in various applications such as finance, physics, and computer science. In this section, we will explore the properties of geometric sequences and series, including how to find the n th term and the sum of a finite and infinite geometric series.
The question
Find the 7 th term of a geometric sequence where the first term is 6 and the common ratio 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 geometric sequences and series, marks are typically awarded for correctly identifying the values of a and r, applying the correct formula, and performing accurate calculations. Partial credit may be given for showing the correct method but making minor arithmetic errors.
What the command words demand
- Find
- Calculate or determine the value of a specific term or sum in a geometric sequence or series.
- Prove
- Show that a given statement about a geometric sequence or series is true using algebraic manipulation and the relevant formulas.
- Determine
- Identify whether a given geometric series converges or diverges, and if it converges, find its sum to infinity.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 3-5 minutes per question to ensure you have enough time to identify the values of a and r, apply the correct formula, and perform necessary calculations accurately.
- Identify the values: a = 6, r = -3, n = 70 marks
- Use the formula for the n th term: a n = ar n-11 mark
- a 7 = 6 × (-3) 7-1 = 6 × (-3) 6 = 6 × 729 = 43742 marks
Final answer: 4374
Work through every step correctly and you earn all 3 marks.
Another worked example
Find the sum of the first 8 terms of a geometric series where the first term is 5 and the common ratio is 2.
- Identify the values: a = 5, r = 2, n = 81 mark
- Use the formula for the sum of the first n terms: S n = a(1 - r n ) / (1 - r)1 mark
- S 8 = 5(1 - 2 8 ) / (1 - 2) = 5(1 - 256) / (-1) = 5(-255) / (-1) = 12752 marks
Final answer: 1275
Work through every step correctly and you earn all 4 marks.
Common mistakes
Using the wrong formula for the sum of a geometric series when r = 1.
Why it happens: Students sometimes forget that the formula S n = a(1 - r n ) / (1 - r) is only valid when r ≠ 1. When r = 1, each term in the series is the same, and the sum of the first n terms is simply na.
Fix: Always check if r = 1 before applying the formula for the sum of a geometric series. If r = 1, use S n = na instead.
Forgetting to check if |r| < 1 when finding the sum to infinity.
Why it happens: Students often forget that the formula for the sum to infinity, S ∞ = a / (1 - r) , is only valid if the series converges. A geometric series converges if and only if |r| < 1.
Fix: Always check if |r| < 1 before applying the formula for the sum to infinity. If |r| ≥ 1, the series does not converge, and the sum to infinity is undefined.
Using the wrong exponent in the n th term formula.
Why it happens: Students sometimes use a n = ar n instead of a n = ar n-1 . This error can lead to incorrect results, especially when calculating the first few terms.
Fix: Always use the correct formula for the n th term: a n = ar n-1 . Double-check your calculations to ensure you are using the right exponent.
Forgetting to simplify fractions in the sum to infinity formula.
Why it happens: Students sometimes leave their answers as fractions without simplifying them, which can lead to incorrect or incomplete answers.
Fix: Always simplify your final answer when using the sum to infinity formula. For example, S ∞ = 8 / (1 - 0.25) = 8 / 0.75 = 32/3 ≈ 10.67 should be simplified to a decimal or fraction as required.
Using the wrong sign for negative common ratios.
Why it happens: Students sometimes make errors when dealing with negative common ratios, especially when raising them to powers. For example, (-2) 3 = -8, not 8.
Fix: Always be careful when working with negative common ratios. Double-check your calculations to ensure you are using the correct sign for each term.
Forgetting to apply the modulus notation correctly.
Why it happens: Students sometimes write |r| < 1 as r < 1, which is incorrect. The condition for convergence is that the absolute value of r must be less than 1.
Fix: Always use the correct modulus notation: |r| < 1. This means -1 < r < 1.
Using the wrong formula for the sum of a finite geometric series.
Why it happens: Students sometimes confuse the formula for the sum of a finite geometric series with other formulas, such as the sum to infinity or the n th term formula.
Fix: Always use the correct formula for the sum of the first n terms: S n = a(1 - r n ) / (1 - r) . Double-check your calculations to ensure you are using the right formula.
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 |
|---|---|---|
| Geometric Sequence Term | Find a specified term of a geometric sequence using the first term and common ratio. | 3 |
| Geometric Series Sum | Find the sum of the first eight terms of a geometric series given its first term and ratio. | 4 |
| Sum To Infinity | Use the formula for the sum to infinity of a convergent geometric series. | 3 |
| Geometric Series Sum | Find the sum of the first five terms of a geometric series using the sum formula. | 4 |
| Sum To Infinity | Find the sum to infinity of a convergent geometric series using the given first term and ratio. | 3 |
| Total across these question types | 17 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.