Algebra

Overview

Sequences are a fundamental topic in GCSE Mathematics that bridge the gap between simple number patterns and complex algebra. A sequence is simply an ordered list of numbers, but the power lies in finding the rule that governs it. Examiners love sequences because they test multiple skills at once: pattern recognition, algebraic notation, and problem-solving. This topic connects heavily to linear graphs (where the common difference is the gradient) and functions.

In your exam, you can expect a range of question styles. Foundation tier often focuses on generating terms from a rule or identifying special sequences. Higher tier demands finding the nth term of quadratic sequences or proving whether a specific number belongs to a sequence. Let's break it all down.

Key Concepts

Concept 1: Term-to-Term Rules

A term-to-term rule tells you how to get from one number in the sequence to the very next number. You must know the previous term to find the next one.

Example: The sequence 3, 7, 11, 15... has a term-to-term rule of "add 4".

This is simple but limited. If an examiner asks for the 100th term, using a term-to-term rule would take forever! This is why we need position-to-term rules.

Concept 2: Position-to-Term Rules (The nth Term)

The "nth term" is an algebraic formula that links the position of a number (n) to its actual value.

Example: If the nth term is 3n + 2:

• The 1st term (n=1) is 3(1) + 2 = 5
• The 10th term (n=10) is 3(10) + 2 = 32

This is incredibly powerful because it allows you to calculate any term instantly without knowing the previous ones.

Concept 3: Special Sequences

Examiners expect you to instantly recognise certain famous sequences. Committing these to memory is an easy way to secure marks.

• Square Numbers: 1, 4, 9, 16, 25... (n^2)
• Cube Numbers: 1, 8, 27, 64, 125... (n^3)
• Triangular Numbers: 1, 3, 6, 10, 15... (Add 2, then 3, then 4...)
• Fibonacci Sequence: 1, 1, 2, 3, 5, 8... (Add the two previous terms together)

Concept 4: Linear (Arithmetic) Sequences

A linear sequence increases or decreases by the same amount every time. This constant amount is called the "common difference".

To find the nth term:

1. Find the common difference (this is the number in front of n).
2. Write out the times table for that number.
3. Find what you need to add or subtract to get to your sequence.

Concept 5: Quadratic Sequences (Higher Tier)

Quadratic sequences have an n^2 in their rule. The key feature is that the first differences are not equal, but the second differences are constant.

To find the nth term:

1. Find the first differences, then the second differences.
2. Halve the second difference. This is the coefficient of n^2.
3. Subtract the n^2 sequence from your original sequence.
4. Find the linear nth term of the remaining numbers.
5. Combine them.

Mathematical Relationships

• Linear nth term: an + b (where a is the common difference)
• Quadratic nth term: an^2 + bn + c (where 2a is the second difference)
• Geometric sequences: ar^{n-1} (where r is the common ratio)

Listen to the Podcast

Review these concepts on the go with our audio guide: