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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Identify the values: a = 6, r = -3, n = 7
- S2
Use the formula for the n th term: a n = ar n-1
- S3
a 7 = 6 × (-3) 7-1 = 6 × (-3) 6 = 6 × 729 = 4374
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Identify the values: a = 6, r = -3, n = 7
- S2
Use the formula for the n th term: a n = ar n-1
- S3
a 7 = 6 × (-3) 7-1 = 6 × (-3) 6 = 6 × 729 = 4374
Final answer: 4374
Common mistakes
- Using the wrong formula for the sum of a geometric series when r = 1. — 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. — 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. — 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. — 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. — 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. — 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. — 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
- Full worked solution (all marking points)3 marks