Sequences and Proportion

This guide covers subtopics 2.7 Sequences and 2.8 Proportion, from Topic 2, Algebra and graphs, for Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series.

Where this fits in 4024

Both subtopics are about recognising a rule connecting quantities — between consecutive terms of a sequence, or between two related variables — and expressing that rule algebraically. They’re grouped here as two compact, pattern-based subtopics that sit between the equation-solving of Inequalities and the graph work later in Topic 2.

Syllabus coverage

CAMBRIDGE O LEVEL MATHEMATICS (SYLLABUS D) 4024

• Continue a given number sequence or pattern (2.7)
• Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences (2.7)
• Find and use the nth term of sequences, including linear, quadratic, cubic and exponential sequences and simple combinations of these; subscript notation may be used, e.g. Tₙ is the nth term of sequence T (2.7)
• Express direct and inverse proportion in algebraic terms, and use this to find unknown quantities, including linear, square, square root, cube and cube root proportion; knowledge of the proportional symbol (∝) is required (2.8)

4024 is not tiered — every candidate covers all of the above.

Sequences

Continuing a sequence or spotting its pattern starts with the term-to-term rule — how each term relates to the one before it (add a constant, multiply by a constant, and so on) — but 4024 also expects recognising relationships between different sequences, such as one sequence being the sequence of differences of another.

The nth term gives a formula for any term directly, without needing every term before it. The type of formula depends on the sequence:

Sequence type	nth term pattern	Example
Linear	Tₙ = an + b	3, 7, 11, 15, … → Tₙ = 4n − 1
Quadratic	Tₙ = an² + bn + c	1, 4, 9, 16, … → Tₙ = n²
Cubic	Tₙ = an³ + …	1, 8, 27, 64, … → Tₙ = n³
Exponential	Tₙ = a × rⁿ⁻¹	3, 6, 12, 24, … → Tₙ = 3 × 2ⁿ⁻¹

Worked example. Find the nth term of the linear sequence 5, 8, 11, 14, …

common difference = 3, so Tₙ = 3n + c
when n = 1, T₁ = 5, so 3(1) + c = 5, c = 2
Tₙ = 3n + 2

Worked example. Find the nth term of the sequence 2, 6, 18, 54, …

each term is 3× the previous term, so this is exponential: Tₙ = a × 3ⁿ⁻¹
when n = 1, T₁ = 2, so a = 2
Tₙ = 2 × 3ⁿ⁻¹

The syllabus also expects simple combinations of these types — for instance a sequence whose nth term is a linear term plus a quadratic term.

Direct and inverse proportion

Two quantities are in direct proportion if one is a constant multiple of the other — as one increases, the other increases at the same rate. Using the proportional symbol ∝:

y ∝ x        means        y = kx        (k a constant)

4024 extends this beyond simple linear proportion to square, square root, cube and cube root proportion:

y ∝ x²        y = kx²
y ∝ √x        y = k√x
y ∝ x³        y = kx³
y ∝ ³√x       y = k³√x

Inverse proportion means one quantity increases as the other decreases, in a constant-product relationship:

y ∝ 1/x        means        y = k/x

Worked example. y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.

y = kx²
45 = k(3²) = 9k
k = 5
y = 5x²
when x = 5: y = 5(25) = 125

Worked example. y is inversely proportional to x. When x = 4, y = 6. Find y when x = 8.

y = k/x
6 = k/4
k = 24
y = 24/x
when x = 8: y = 24/8 = 3

The method is the same in every case: use the given pair of values to find the constant k, then use k to find the unknown quantity.

Common mistakes

• Assuming every sequence is linear. Check the differences between terms first — if they’re not constant, check second differences (constant second differences indicate a quadratic sequence) before assuming a more complex form.
• Confusing direct and inverse proportion. In direct proportion, both quantities increase together; in inverse proportion, one increases as the other decreases — writing y = kx when the relationship is actually y = k/x (or vice versa) is a common and costly error.
• Forgetting to find k first. Every proportion problem requires finding the constant from the given pair of values before it can be used to find anything else.
• Using x instead of x², √x, x³ or ³√x — always check exactly which power or root the question specifies before setting up the proportion equation.

Quick revision checklist

• Term-to-term rules and recognising relationships between sequences
• Finding the nth term for linear, quadratic, cubic and exponential sequences, and simple combinations
• Setting up y = kx (or the square/root/cube variant) from y ∝ x, finding k, then solving
• Direct vs inverse proportion, and not confusing the two

Related resources

• Inequalities — the previous subtopic
• Graphs in Practical Situations — the next subtopic
• Cambridge O Level Mathematics subject hub

Written against Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series. Always check the current syllabus for your examination year.