Sequences

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Continue a given number sequence or pattern
  2. Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences
  3. Find and use the nth term of the following sequences:
    • (a) linear sequences
    • (b) simple quadratic sequences
    • (c) simple cubic sequences
  4. Work with examples like: write the next two terms in 1, 3, 6, 10, 15, ... and find the nth term of 2, 5, 10, 17
  5. Find and use the nth term of sequences
  6. Use subscript notation (Tn for the nth term) and work with linear, quadratic, cubic, exponential sequences and simple combinations of these

What is a Sequence?

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A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in the sequence is called a term.

For example:

  • 3, 6, 9, 12, 15, ... (the rule is "add 3 each time")
  • 1, 4, 9, 16, 25, ... (these are square numbers)

Position and Terms

Every term in a sequence has a position. We use the letter n to represent the position number:

  • When n = 1, we're talking about the 1st term
  • When n = 2, we're talking about the 2nd term
  • When n = 3, we're talking about the 3rd term
  • When we don't know the position, we call it the nth term

Subscript Notation

Another way to write terms uses subscript notation:

  • a₁ means the 1st term
  • a₂ means the 2nd term
  • a₃ means the 3rd term
  • aₙ means the nth term (the general term)

Similarly, if we have a sequence called T, we can write:

  • T₁ for the 1st term
  • T₂ for the 2nd term
  • Tₙ for the nth term

Example: In the sequence 4, 5, 6, 7, 8, ...

  • a₁ = 4 (the 1st term is 4)
  • a₂ = 5 (the 2nd term is 5)
  • a₄ = 7 (the 4th term is 7)

Continuing Sequences

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To continue a sequence means to find the next terms by spotting the pattern.

Method 1: Look at First Differences

The first differences are what you add (or subtract) to get from one term to the next.

Example 1: 4, 7, 10, 13, ...

  • 4 → 7: add 3
  • 7 → 10: add 3
  • 10 → 13: add 3

The first differences are all +3, so the next term is 13 + 3 = 16.

Example 2: 20, 18, 16, 14, ...

  • 20 → 18: subtract 2
  • 18 → 16: subtract 2
  • 16 → 14: subtract 2

The first differences are all -2, so the next term is 14 - 2 = 12.

Method 2: Look for Patterns in the Differences

Sometimes the differences themselves follow a pattern.

Example 3: 2, 8, 15, 23, ...

  • 2 → 8: add 6
  • 8 → 15: add 7
  • 15 → 23: add 8

The differences are 6, 7, 8 (increasing by 1 each time). So the next difference is 9, and the next term is 23 + 9 = 32.

Example 4: 1, 3, 7, 15, ...

  • 1 → 3: add 2
  • 3 → 7: add 4
  • 7 → 15: add 8

The differences are 2, 4, 8 (doubling each time). So the next difference is 16, and the next term is 15 + 16 = 31.

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