Quadratic sequences · Higher notes

A quadratic sequence has a constant second difference. Its nth term has the form an² + bn + c, with counting starting at n = 1 unless stated otherwise. The second difference is 2a, so halve it to identify the square-term coefficient. Subtract an² from each original term: what remains is a linear sequence whose nth term you already know how to find. Combine the two parts, then test the rule using several positions rather than only the first.

Learning objectives

Find a quadratic rule from first and second differences.
Use the rule to calculate a term at an integer position.

Key ideas

First differences compare consecutive terms; second differences compare consecutive first differences.

A quadratic sequence has a constant second difference twice its quadratic coefficient.

T_n=an^2+bn+c,\qquad\text{second difference}=2a

Subtract the square part; find a linear rule for the remainder.

The first term is position one unless another start is specified.

n\in\{1,2,3,\ldots\}

Worked examples

Example 1

Find the rule for the sequence.

3,8,15,24,35,\ldots

\text{first differences}:5,7,9,11;\quad\text{second differences}:2,2,2

Subtract the square-number sequence to obtain a linear remainder.

a=\frac22=1;\qquad\text{remainder}:2,4,6,8,10,\ldots

T_n-n^2=2n

Answer

T_n=n^2+2n

Example 2

Find the fourth term of the quadratic sequence.

T_n=2n^2-n,\qquad T_4=2(4^2)-4

Answer

T_4=28

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Find the nth term of a quadratic sequence using differences.

AO2 · Reasoning

Explain why a proposed rule fits every term rather than only the first terms.

AO3 · Problem solving

Model a growing pattern and determine a later stage or a possible stage number.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

The quadratic coefficient is half the second difference. Check the complete rule against several original terms.

Try it yourself

Attempt each question before opening its answer.

Check 1: For nth term 2n² - n, find the fourth term.

Answer: 28

Substitute n = 4: 2 × 16 - 4 = 28.

Check 2: A quadratic sequence has second difference 8. What is the n² coefficient?

Answer: 4

The coefficient is half the second difference.

Explain it in your own words

Write how to find first and second differences.
Explain why you halve the second difference.
Find a rule for 2, 7, 14, 23, ... and check its first two terms.

Keep practising

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Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.