Sequences
A sequence is an ordered list of numbers that follows a specific pattern or rule. In arithmetic sequences, each term increases or decreases by the same fixed amount called the common difference. For example, the sequence 3, 7, 11, 15 has a common difference of 4, making it an arithmetic sequence covered in Year 7 of the UK National Curriculum.
Why it matters
Sequences appear throughout mathematics and real-world applications. In finance, arithmetic sequences model regular savings plans where someone deposits £20 monthly, creating the sequence £20, £40, £60, £80. Geometric sequences describe population growth, where a bacteria culture might double every hour: 100, 200, 400, 800 cells. Construction projects use sequences for scheduling tasks and calculating materials. The nth term formula becomes essential for GCSE algebra, whilst sequence patterns underpin calculus series at A-level. Understanding sequences develops logical reasoning skills needed for computer programming, where algorithms often process data in sequential patterns. Estate agents use arithmetic sequences to calculate mortgage payments, and scientists apply geometric sequences to model radioactive decay with half-lives.
How to solve sequences
Sequences
- Arithmetic sequence: constant difference (d) between terms. aₙ = a₁ + (n−1)d.
- Geometric sequence: constant ratio (r) between terms. aₙ = a₁ × rn−1.
- To identify: check differences first, then ratios.
- Sum of arithmetic series: S = n/2 × (first + last).
Example: 2, 6, 18, 54: ratio = 3, geometric. a₅ = 2 × 3⁴ = 162.
Worked examples
Write the next 3 terms: 8, 11, 14, __, __, __
Answer: 17, 20, 23
- Find the common difference → d = 3 — 11 − 8 = 3. Each term increases by 3.
- Continue the pattern → 17, 20, 23 — 14 + 3 = 17, 17 + 3 = 20, 20 + 3 = 23.
Find the 10th term of: 4, 10, 16, 22, ...
Answer: 58
- Identify first term and common difference → a₁ = 4, d = 6 — First term is 4. Difference: 10 − 4 = 6.
- Use the nth term formula → aₙ = a₁ + (n − 1)d — The nth term of an arithmetic sequence is a₁ + (n − 1)d.
- Substitute → a_10 = 4 + (10 − 1) × 6 — Replace a₁ with 4, n with 10, d with 6.
- Calculate → 58 — 4 + 9 × 6 = 4 + 54 = 58.
Find the common difference and the 20th term: 1, 4, 7, 10, ...
Answer: d = 3, 20th term = 58
- Find the common difference → d = 4 − 1 = 3 — Subtract consecutive terms: 4 − 1 = 3.
- Use the nth term formula → a₂₀ = 1 + (20 − 1) × 3 — aₙ = a₁ + (n − 1)d with n = 20.
- Calculate → 58 — 1 + 19 × 3 = 1 + 57 = 58.
Common mistakes
- Confusing arithmetic and geometric patterns leads to writing the sequence 2, 6, 18, 54 as having a common difference of 4 instead of recognising the common ratio of 3.
- Incorrectly applying the nth term formula produces errors like calculating the 5th term of 3, 7, 11, 15 as 3 + 5 × 4 = 23 instead of using 3 + (5-1) × 4 = 19.
- Adding the common difference incorrectly gives wrong terms, such as continuing 8, 11, 14 as 8, 11, 14, 16, 18 instead of 8, 11, 14, 17, 20.