GCSE MATHS • HIGHER TIER

Quadratic Sequences Made Simple

Use first and second differences to identify quadratic patterns, form nth-term rules and find specific terms.

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Quadratic Sequences in Four Moves

1

Find First Differences

Subtract consecutive terms to see how the sequence changes.

2

Find Second Differences

Calculate the differences between the first differences.

3

Form the Rule

Use the constant second difference to build an nth-term rule in the form an² + bn + c.

4

Check and Substitute

Check the rule against the sequence and substitute n to find specific terms.

What You’ll Learn

Understand what a quadratic sequence is.

Calculate first and second differences.

Distinguish arithmetic and quadratic sequences.

Recognise the general form an² + bn + c.

Find quadratic nth-term rules.

Use nth-term rules to calculate specific terms.

Key Rules

First Differences

Subtract consecutive terms to find how the sequence changes.

Second Differences

Find the differences between the first differences

Quadratic nth Term

A quadratic sequence usually has an nth-term rule in the form an² + bn + c.

Coefficient of n²

Divide the constant second difference by 2 to find the value of a.

Worked Proof: Quadratic Sequences

Step-by-Step Method

  • 1. Calculate first differences.
  • 2. Calculate second differences.
  • 3. Divide the second difference by 2 to find a.
  • 4. Build the n² term.
  • 5. Compare with the sequence.
  • 6. Adjust using bn and c.

More Worked Examples

1

Visual nth Term Structure

Quadratic nth term: an² + bn + c

Find the nth term: 1, 4, 9, 16, 25
Sequence of square numbers.
Answer: n²

2

Is this quadratic? 3, 7, 11, 15

First Differences: 4, 4, 4

Answer: No

3

Find the nth term: 3, 8, 15, 24, 35

Second Difference = 2

Coefficient of n² = 1
Rule: n² + 2n

4

Is this quadratic? 2, 6, 12, 20, 30

First Differences: 4, 6, 8, 10

Second Differences: 2, 2, 2
Answer: Yes
  • Common GCSE Mistakes
  • Forgetting to calculate second differences.
  • Using arithmetic methods on quadratic sequences.
  • Using an incorrect coefficient for n².
  • Making substitution errors.
  • Making arithmetic mistakes.
  • Exam Tips
  • Always calculate both first and second differences.
  • Constant first differences indicate an arithmetic sequence, not a quadratic one.
  • Divide the constant second difference by 2 to find the coefficient of n².
  • Check your nth-term rule against several terms.
  • Substitute the term position carefully when finding a specific term.
  • Use a difference table to keep your working organised.

Skills Used in Quadratic Sequences

Quadratic Sequences

First Differences

Second Differences

nth Term

n² Patterns

Substitution

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Frequently Asked Questions

A quadratic sequence has constant second differences and its nth term contains an n² term.

Arithmetic sequences have constant first differences, while quadratic sequences have constant second differences.

It is usually written in the form an² + bn + c.

The guide's method says to divide the constant second difference by 2.

Substitute the term number for n in the nth-term rule.

They make the changing first differences and constant second differences easier to identify.