Linear Sequences Worksheets
Continuing Sequences from Patterns
Year groups: 7
Finding Arithmetic nth Terms Worksheet
Year groups: 7, 8
Finding nth Terms from Patterns
Year groups: 7, 8
Generating Sequences from the Nth Term
Year groups: 7, 8
Generating Sequences from the Term to Term Rule
Year groups: 7, 8
Using the Nth Term (Linear)
Year groups: 8, 9
Iterative Notation and Arithmetic Sequences
Year groups: 10, 11
What is a linear sequence in maths?
A linear sequence is a pattern of numbers where the difference between consecutive terms remains constant. This fixed difference is called the common difference. For example, in the sequence 3, 7, 11, 15, the common difference is 4. Linear sequences are also known as arithmetic sequences or arithmetic progressions.
The nth term formula for a linear sequence takes the form an + b, where 'a' represents the common difference and 'b' is a constant that determines the sequence's starting position. Understanding linear sequences forms the foundation for more advanced algebra topics, including quadratic sequences and geometric progressions. Students typically encounter this topic in Year 7 and continue developing their skills through to GCSE.
Which year groups study linear sequences?
Linear sequences are introduced in Year 7 as part of the Key Stage 3 algebra curriculum and continue through Year 8 and Year 9. The topic then reappears at GCSE level in Years 10 and 11, where students tackle more complex problems involving nth term derivation and proof.
At KS3, students focus on recognising patterns, finding the next terms, and generating simple nth term formulae. By KS4, the expectations increase to include justifying whether given numbers belong to specific sequences, solving equations involving nth terms, and applying sequence knowledge to problem-solving contexts. Our worksheets are differentiated to match these progression expectations across all year groups.
How do you find the nth term of a linear sequence?
To find the nth term of a linear sequence, first identify the common difference by subtracting consecutive terms. This becomes the coefficient of n in your formula. Next, work out what to add or subtract by comparing the sequence values with multiples of the common difference.
For instance, with the sequence 5, 8, 11, 14, the common difference is 3, giving 3n. Comparing with 3n (which produces 3, 6, 9, 12), you need to add 2 to match the original sequence, giving the nth term as 3n + 2. Our worksheets provide extensive practice with this method, starting with straightforward examples and progressing to sequences with negative common differences and larger numbers.
Do your linear sequences worksheets include answers?
Yes, every linear sequences worksheet includes a complete answer sheet. This allows students to check their working independently, promoting self-assessment and helping them identify areas where they need additional support. Teachers can use the answer sheets to mark efficiently or provide them to students for peer marking activities.
All worksheets are available as downloadable PDFs, making them straightforward to print for classroom use or share digitally for remote learning. The answer sheets show final answers clearly, enabling quick checking whilst students develop their understanding of nth terms, common differences, and sequence generation. This combination of practice materials and solutions supports both independent study and structured classroom teaching.