AQA A-Level Mathematics Paper 1, June 2025: Question 6

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

6 n 250

Substitutes n = 250 in and S250 = (2×3 +(250 −1)×−0.5)

(n −1) = −14812.5

and at least one of

a = 3 or d = −0.5 into

n ( )

2a +(n −1)d

2 1.1a M1

Or into a + (n −1)d and

n

(a + l)note the correct

value of l = -121.5

Or into a + (n −1)d

Condone poor bracketing

29625

Obtains –14812.5 or −

2 1.1b A1

Question 6 Total 2

How to answer it

Sum of an Arithmetic Series

📋 What this question tests

This question tests your ability to recall and apply the formula for the sum of an arithmetic progression (AP), correctly substituting given parameters ( a , d , and n ), with careful handling of negative signs during algebraic arithmetic.

Question Analysis & Solution

AQA A-Level Pure Mathematics • 2 Marks

💡 Key Knowledge

First term: a = 3
Common difference: d = −0.5
Number of terms: n = 250
Sum formula:
S_n = (n / 2)[2a + (n − 1)d]
Alternative approach:
l = a + (n − 1)d

S_n = (n / 2)(a + l)

✅ Correct Answer

The sum of the first 250 terms is:

−14812.5 or −29625 / 2

Mark Scheme Breakdown:
• M1 (AO 1.1a): Correct substitution of n = 250 and at least one of a = 3 or d = −0.5 into the sum formula.
• A1 (AO 1.1b): Correct exact final value ( −14812.5 or equivalent fraction).

📐 Step-by-Step Calculations

Method 1: Direct Substitution into S_n Formula

Identify values:
a = 3 , d = −0.5 , n = 250
Substitute into the sum formula:
S₂₅₀ = (250 / 2) × [2(3) + (250 − 1)(−0.5)] [M1 Awarded]
Simplify the bracketed terms:
S₂₅₀ = 125 × [6 + 249 × (−0.5)]
S₂₅₀ = 125 × [6 − 124.5]
S₂₅₀ = 125 × (−118.5)
Evaluate final product:
S₂₅₀ = −14812.5 (or −29625/2 ) [A1 Awarded]

Method 2: Finding the Last Term (l) First

Calculate the 250th term (l):
u₂₅₀ = a + (n − 1)d = 3 + 249(−0.5) = 3 − 124.5 = −121.5
Use S_n = (n / 2)(a + l):
S₂₅₀ = (250 / 2)(3 + (−121.5)) = 125 × (−118.5) = −14812.5

❌ Common Errors & Traps

Sign errors with d: Writing +0.5 instead of −0.5 , which gives an incorrect positive total (+15562.5).
Missing brackets on negative multiplier: Writing 249 − 0.5 instead of 249 × (−0.5) on calculators, leading to incorrect subtraction.
Confusing 'a' with '2a': Writing 3 instead of 2(3) = 6 inside the main formula bracket.
Off-by-one term error: Using n instead of n − 1 , multiplying by 250 instead of 249 .

🧠 Exam Technique & Examiner Tips

Show substituted formula: Always write out the unsimplified substitution line. Even if a typing error occurs on the calculator, writing S₂₅₀ = (250/2)[2(3) + (249)(-0.5)] locks in the M1 method mark.
Sense check the result: The first term is 3, but subtracting 0.5 each time means nearly all 250 terms are large negative numbers. The overall sum must be negative.
Formula booklet reference: Double-check the Pure Mathematics formula sheet if you ever hesitate between arithmetic S_n = (n/2)[2a + (n-1)d] and geometric S_n = a(1 - rⁿ)/(1 - r) .

Topics

Pure Mathematics · D: Sequences and series