Sequences form a foundational element of the GCSE Maths curriculum and frequently appear in both foundation and higher-tier papers. They are essential for reasoning, identifying patterns, and developing algebraic expressions. Whether you’re analysing a linear arithmetic pattern, identifying a geometric progression, or exploring the structure of a quadratic sequence, mastering sequences equips students with essential problem-solving and algebraic thinking skills. This comprehensive guide will walk you through everything from basic arithmetic series to more complex quadratic patterns, offering step-by-step examples and exam-style tips along the way.

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🔹 What is a Sequence?

A sequence is an ordered list of numbers that follow a particular rule or formula. Each number in a sequence is called a term, and the rule used to move from one term to the next is often defined either by a recursive (term-to-term) rule or a direct (position-to-term) algebraic rule. Understanding the relationship between terms enables you to predict missing values, create general formulas, and explain number patterns mathematically.

✅ Common Types of Sequences:

• Arithmetic (Linear) – where each term increases or decreases by a fixed amount
• Geometric – where each term is multiplied or divided by a constant factor
• Quadratic – where the sequence grows based on a squared term
• Fibonacci-type or Recurring Sequences – built using previous terms

🔸 Arithmetic Sequences

An arithmetic sequence has a common difference between each term. It is the simplest form of numerical pattern and commonly tested in both foundation and higher-level questions. Recognising the difference helps you develop a general rule (nth term) and solve problems involving term values and positions.

📌 Formula for the nth Term:

nth term = a + (n − 1)d

Where:

• a = first term of the sequence
• d = common difference between terms

🧮 Example:

Sequence: 3, 7, 11, 15, …

• a = 3, d = 4
• nth term = 3 + (n − 1) × 4 = 4n − 1

You can now find any term in the sequence. For instance, the 10th term would be:

4 × 10 − 1 = 39

🔸 Geometric Sequences

Geometric sequences grow by multiplying or dividing by the same constant ratio. While less common in GCSE exams, they may appear in challenging problem-solving questions, especially in topics involving growth, decay, or proportional reasoning.

📌 Formula for the nth Term:

nth term = a × rⁿ⁻¹

Where:

• a = first term
• r = common ratio between terms

🧮 Example:

Sequence: 2, 4, 8, 16, …

• a = 2, r = 2
• nth term = 2 × 2ⁿ⁻¹

To find the 5th term:

2 × 2⁴ = 2 × 16 = 32

Note: Always check whether the pattern multiplies/divides consistently before applying the geometric rule.

🔸 Quadratic Sequences

Quadratic sequences grow at a changing rate, typically represented by a squared term. You’ll notice that the second differences between terms remain constant, rather than the first. This type of sequence requires forming and solving a quadratic expression.

📌 General Form:

nth term = an² + bn + c

Where a, b, and c are constants.

✅ Steps to Find the nth Term:

1. Write the first few terms of the sequence
2. Calculate first and second differences
3. Use second difference to find a
4. Use substitution and simultaneous equations to find b and c

🧮 Example:

Sequence: 2, 6, 12, 20, 30

• First differences: 4, 6, 8, 10
• Second difference = 2 ⇒ a = 1
• Plug into nth term formula: n² + bn + c
• Use n = 1: 1 + b + c = 2
• Use n = 2: 4 + 2b + c = 6
• Solve the simultaneous equations:

b = 0, c = 1 → nth term = n² + 1

🔸 Special Sequences

✅ Fibonacci Sequence:

In this famous sequence, each term is the sum of the previous two:

1, 1, 2, 3, 5, 8, 13, …

Used in real-life applications like biology and finance, it is occasionally tested for its rule-forming logic.

✅ Triangle Numbers:

These represent items arranged in an equilateral triangle shape:

1, 3, 6, 10, 15, …

Formula: n(n + 1)/2

Often asked in pattern-based or problem-solving tasks.

🧠 Exam Tips for Sequences

• Check for a common first or second difference
• Identify whether the rule is arithmetic, geometric, or quadratic
• Use structured steps to find the nth term, especially for quadratic cases
• Double-check your rule by substituting the first 2–3 terms
• Look for misleading patterns or exceptions in worded questions

📝 Practice Questions

1. Find the nth term of the sequence: 5, 10, 15, 20, …
2. What is the 12th term of: 3, 6, 12, 24, …?
3. Determine whether the sequence 2, 6, 12, 20, 30 is quadratic and find its nth term.
4. List the first 5 triangle numbers and write their formula.
5. A sequence has the nth term: 3n² − 2n + 5. Find the first 3 terms.
6. Identify the type and rule for the sequence: 1, 4, 9, 16, 25
7. Explain why the sequence: 8, 6, 4, 2, 0 is arithmetic and find its rule.

✅ Final Thoughts

Understanding sequences and patterns not only helps in exam situations but also strengthens your logical thinking and algebraic fluency. By identifying term rules and forming nth term expressions accurately, you can confidently tackle structured questions and word problems. Always practice applying these principles to both numerical sequences and contextual scenarios.

For more support, interactive lessons, downloadable resources, and one-on-one help, visit GCSE Maths Tutor.