GCSE Mathematics – Higher Tier
Sequences
Quadratic Sequences
Introduction to Finding the Nth Term of Quadratic Sequences
Lesson Duration: 60 Minutes
Curriculum: AQA GCSE Mathematics / Pearson Edexcel GCSE Mathematics
In this lesson, students learn how to determine the nth term of quadratic sequences by identifying the first and second differences before constructing the general expression in the form:
$$
a_n = an^2 + bn + c
$$
Students use a structured step-by-step approach to determine the values of a, b and c, before verifying their solution by substituting values of n. Through teacher modelling, guided practice and exam-style questions, students develop confidence in recognising quadratic patterns and solving increasingly complex sequence problems.
To determine the nth term of quadratic sequences using first differences, second differences and algebraic reasoning.
By the end of this lesson students will be able to:
✓ Identify quadratic sequences.
✓ Calculate first differences.
✓ Calculate second differences.
✓ Determine the values of a, b and c.
✓ Write the nth term.
✓ Verify the rule by substitution.
Quadratic sequences appear throughout GCSE Mathematics and are closely linked to:
Algebra
Graphs
Quadratic Functions
Modelling
Problem Solving
Understanding quadratic sequences develops algebraic reasoning and prepares students for higher-level mathematics.
What is meant by the nth term of a sequence?
Find the nth term of:
2, 5, 8, 11, …
Find the first differences of
6, 11, 16, 21
Explain why a constant first difference indicates a linear sequence.
Sequence
Pattern
Term
Difference
First Difference
Second Difference
Quadratic
General Term
Coefficient
Substitute
Display the sequence:
7, 15, 29, 49, …
Ask students:
What do you notice?
Is the sequence increasing?
Are the first differences constant?
What happens when we calculate the second differences?
Allow discussion before introducing the formal method.
Students complete the Frayer Model.
A quadratic sequence is a sequence in which the second differences are constant.
Constant second difference.
General term is:
$$
an^2+bn+c
$$
Produces a parabolic graph.
7,15,29,49,…
3,8,15,24,…
2,5,8,11,…
5,10,20,40,…
Using the example shown.
7,15,29,49,…
Find the first differences.
8
14
20
Find the second differences.
6
6
Since the second difference is constant, the sequence is quadratic.
Find a
$$
2a=6
$$
Therefore
$$
a=3
$$
Find b
$$
3a+b=8
$$
$$
9+b=8
$$
$$
b=-1
$$
Find c
Use the first term.
$$
a+b+c=7
$$
$$
3-1+c=7
$$
$$
2+c=7
$$
$$
c=5
$$
Write the nth term.
$$
\boxed{3n^2 – n + 5}
$$
Check
Substitute
n=1
n=2
n=3
Confirm the sequence.
Complete together.
4,11,24,43,…
9,19,35,57,…
6,17,34,57,…
Teacher questions throughout.
Questions 1–4
Questions 5–8
Questions 9–12
Students explain every step using full mathematical reasoning.
Students often:
❌ Use the first difference instead of the second difference.
❌ Forget that
$$
2a=\text{second difference}
$$
❌ Use the wrong equation for b.
❌ Forget to verify the nth term.
Teacher models each misconception and correction.
Always check your nth term by substituting the first three values of n.
If your answers do not match the original sequence, revisit your calculations.
The sequence begins
12
25
44
69
Find the nth term.
Explain every step.
Students complete an Exit Ticket.
How do you know a sequence is quadratic?
Why is
$$
2a=\text{second difference}
$$
Write one thing you found difficult today.
Complete the Tutor Grade Worksheet:
Retrieval Practice
Exam-style Questions
Mixed GCSE Questions
Reflection Activity
Tutor Grade Teacher Video
Tutor Grade Frayer Model™
PowerPoint Presentation
Worked Example Sheet
Guided Practice Worksheet
Independent Practice Worksheet
Homework Sheet
Mark Scheme
Interactive Quiz
Printable Notes
Parent Support Guide
I Do → We Do → You Do → Reflect → Retrieve
This lesson follows the Tutor Grade evidence-informed teaching model, combining retrieval practice, explicit instruction, teacher modelling, guided practice, independent application and formative assessment. The method shown in your image is used as the core instructional sequence, ensuring students understand not only how to find the nth term of a quadratic sequence, but also why each algebraic step is necessary. This provides a consistent lesson structure that can be replicated across all 342 Tutor Grade mathematics lessons.
