GCSE MATHS • HIGHER TIER

Arithmetic Sequences Made Simple

Recognise arithmetic patterns, find common differences, form nth term rules and calculate specific terms.

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Worked proofs

Arithmetic Sequences in Four Moves

1

Compare

Look at consecutive terms in the sequence.

2

Find the Difference

Calculate the common difference and check that it stays constant.

3

Form the Rule

Use the common difference as the coefficient of n and adjust with a constant.

4

Substitute

Use the nth term rule to calculate any required term.

What You’ll Learn

Understand what an arithmetic sequence is.

Find and use the common difference.

Identify whether a sequence is arithmetic.

Continue increasing and decreasing arithmetic sequences.

Find nth term rules.

Use nth term rules to find specific terms.

Key Rules

The nth Term Rule

The nth term allows students to calculate any term in a sequence without listing all previous terms.

Visual nth Term Example

Sequence: 3, 7, 11, 15, 19
Common Difference = 4
nth Term = 4n - 1

Worked Proof: Arc Length

How to Find an nth Term

  • 1. Find the common difference.
  • 2. Use it as the coefficient of n.
  • 3. Compare with the first term.
  • 4. Adjust with a constant.

More Worked Examples

1

Finding Specific Terms

Once the nth term is known, substitute the term number.

nth term = 4n - 1
Find the 20th term.
4(20) - 1 = 80 - 1 = 79

2

Negative Arithmetic Sequences

Arithmetic sequences can decrease as well as increase.

50, 45, 40, 35
Difference = -5
nth term = 55 - 5n

3

Visual nth Term Example

Sequence: 3, 7, 11, 15, 19

Common Difference = 4
nth Term = 4n - 1

4

Visual nth Term Example

Find the nth term: 5, 8, 11, 14, 17

Difference = 3
nth term = 3n + 2
  • Common GCSE Mistakes
  • Calculating the common difference incorrectly.
  • Mixing up arithmetic and geometric sequences.
  • Making errors when forming nth term rules.
  • Making substitution mistakes.
  • Making arithmetic errors.
  • Exam Tips
  • Check that the difference is constant before calling a sequence arithmetic.
  • Use the common difference as the coefficient of n.
  • Compare your n-rule with the first term to find the constant adjustment.
  • Remember that arithmetic sequences can decrease.
  • Check your nth term against several terms in the sequence.

Skills Used in Arc Length

Sequences

Common Difference

nth Term

Algebraic Patterns

Substitution

Negative Numbers

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Exam-Style Questions

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Watch the Arithmetic Sequences Tutorial

Step-by-step walkthroughs of Arithmetic Sequences with clear methods and exam tips.

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Frequently Asked Questions

It is a sequence where the difference between consecutive terms is always the same.

It is the fixed amount added or subtracted between consecutive terms.

Calculate consecutive differences. If they remain constant, the sequence is arithmetic.

Add the common difference to the previous term.

It allows you to calculate any term without listing all previous terms.

Yes. The source gives decreasing sequences with negative common differences.