AQA A-Level Mathematics Paper 1, June 2025: Question 2

1 mark · Easy difficulty · Short Answer

Identify the property describing a given alternating sequence defined by a recurrence relation.

Practise this question

Question

Question 2 gives a sequence defined by the recurrence relation x_{n+1} = -1/4 x_n with x_1 = 32. The first four terms are listed as 32, -8, 2, -0.5. It asks which one of the following terms can be used to describe this sequence: Convergent, Decreasing, Increasing, or Periodic.
Question text

2 A sequence is defined by

– 1

xn+1 = xn with x1 = 32

The first four terms of the sequence are

32, –8, 2, –0.5

Which one of the following can be used to describe this sequence?

Circle your answer.

[1 mark]

Convergent Decreasing Increasing Periodic

Mark scheme

Show the mark scheme Mark scheme table for Question 2 indicates 1 mark (R1, AO 2.2a) for circling the first option, with the typical solution given as 'Convergent'.

Q Marking instructions AO Marks Typical solution

2 Circles first option 2.2a R1 Convergent

Question 2 Total 1

How to answer it

Classifying Sequences: Convergence vs Monotonicity

📌 What this question tests

Understanding sequence definitions and behaviours: recognizing geometric sequences, identifying whether an alternating sequence is convergent, increasing, decreasing, or periodic, and interpreting the effect of a negative common ratio (|r| < 1).

Question 2 Analysis

A-Level Mathematics • Pure • 1 Mark (AO 2.2a)

Question Statement:
A sequence is defined by xn+1 = -(1/4)xn with x1 = 32 .
The first four terms are: 32, -8, 2, -0.5
Which one of the following can be used to describe this sequence?
Options: Convergent  |  Decreasing  |  Increasing  |  Periodic

✅ Correct Answer

Convergent

The first option (Convergent) must be clearly circled to be awarded the reasoning mark R1 .

💡 Key Knowledge

  • Geometric Form: Here, r = -1/4 = -0.25 .
  • Convergence Condition: A geometric sequence converges if and only if -1 < r ≤ 1 . Since |-1/4| < 1 , terms tend to a single finite limit: xn → 0 as n → ∞ .
  • Alternating Signs: Because r < 0 , the signs alternate (+, -, +, -...). This sequence oscillates around zero while getting closer to it.

📐 Step-by-Step Option Elimination

  1. Check "Increasing": Requires xn+1 > xn for all n. Here, x2 (-8) < x1 (32) . False.
  2. Check "Decreasing": Requires xn+1 < xn for all n. Here, x3 (2) > x2 (-8) . False.
  3. Check "Periodic": Requires xn+k = xn for a fixed integer order k (terms repeat identical values). The values change magnitude every term (32, 8, 2, 0.5...). False.
  4. Check "Convergent": As n → ∞ , |xn| → 0 . The terms approach a fixed finite limit ( 0 ). True.

❌ Common Errors & Traps

  • Thinking it is Decreasing: Students notice the absolute values are getting smaller ( 32 → 8 → 2 → 0.5 ) and hastily pick "Decreasing". Remember: a decreasing sequence must satisfy xn+1 ≤ xn across every single step.
  • Confusing Periodic with Alternating: An alternating sign sequence is only periodic if the numbers themselves cycle through identical values (e.g. 2, -2, 2, -2... with r = -1 ).

🧠 Exam Technique & Examiner Commentary

This is an assessment of Assessment Objective AO 2.2a (constructing a rigorous mathematical argument / correct classification).

  • Ensure your circle clearly encloses only Convergent. If you change your mind, cross out previous markings definitively to avoid ambiguity.
  • Always test the strict algebraic inequalities for increasing and decreasing against both positive and negative transitions before choosing.

Topics

Pure Mathematics · D: Sequences and series

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.