Sequences

Cambridge IGCSE Mathematics (0580) · Unit 2: Algebra and graphs · 18 flashcards

Sequences is topic 2.4 in the Cambridge IGCSE Mathematics (0580) syllabus , positioned in Unit 2 — Algebra and graphs , alongside Algebraic notation and manipulation, Equations and Inequalities. In one line: A sequence is an ordered list of numbers (or other elements) called terms. Each term follows a specific pattern or rule.

This topic is examined across Paper 1 (Core) or Paper 2 (Extended) — non-calculator — and Paper 3 (Core) or Paper 4 (Extended) — calculator.

The deck below contains 18 flashcards — 5 definitions and 3 key concepts — covering the precise wording mark schemes reward. Use the 5 definition cards to lock down command-word answers (define, state), then move on to the concept and application cards to handle explain, describe and compare questions.

Key definition

A 'sequence' in mathematics and provide an example

A sequence is an ordered list of numbers (or other elements) called terms. Each term follows a specific pattern or rule.

Example: 2, 4, 6, 8, ... is a sequence of even numbers.

Questions this Sequences deck will help you answer

› What is the 'nth term' of a sequence and how is it useful?
› Write down the first five terms of the Fibonacci sequence and explain the rule.
› How do you find the nth term of a quadratic sequence?

Definition Flip

Define a 'sequence' in mathematics and provide an example.

Answer Flip

A sequence is an ordered list of numbers (or other elements) called terms. Each term follows a specific pattern or rule.

Example: 2, 4, 6, 8, ... is a sequence of even numbers.

Key Concept Flip

What is the 'nth term' of a sequence and how is it useful?

Answer Flip

The 'nth term' is a formula that allows you to calculate any term in the sequence directly based on its position (n). It's useful for finding specific terms without listing all the preceding terms.

Example: For sequence 2n+1, the 5th term is 2(5)+1 = 11.

Key Concept Flip

Find the next two terms in the sequence: 3, 7, 11, 15, ...

Answer Flip

This is an arithmetic sequence with a common difference of 4. The next two terms are 19 (15 + 4) and 23 (19 + 4).

Definition Flip

Explain the difference between an arithmetic and a geometric sequence.

Answer Flip

An arithmetic sequence has a constant difference between consecutive terms (addition/subtraction). A geometric sequence has a constant ratio between consecutive terms (multiplication/division).

Definition Flip

What is the 'common difference' in an arithmetic sequence, and how do you find it?

Answer Flip

The 'common difference' is the constant value added to each term to get the next term in an arithmetic sequence. You find it by subtracting any term from the term that follows it.

Example: In the sequence 1, 4, 7, 10..., the common difference is 4 - 1 = 3.

Key Concept Flip

The 4th term of an arithmetic sequence is 14 and the 7th term is 23. Find the common difference.

Answer Flip

The difference between the 7th and 4th term (23 - 14 = 9) spans 3 common differences. Therefore, the common difference is 9 / 3 = 3.

Definition Flip

What is the 'common ratio' in a geometric sequence, and how do you find it?

Answer Flip

The 'common ratio' is the constant value multiplied by each term to get the next term in a geometric sequence. You find it by dividing any term by the term that precedes it.

Example: In the sequence 2, 6, 18, 54..., the common ratio is 6 / 2 = 3.

Key Concept Flip

Determine the nth term of the following arithmetic sequence: 5, 8, 11, 14, ...

Answer Flip

The common difference is 3. The nth term is of the form 3n + c. Substitute n = 1: 3(1) + c = 5, so c = 2. Therefore, the nth term is 3n + 2.

Definition Flip

Explain what a quadratic sequence is and give an example.

Answer Flip

A quadratic sequence is one where the nth term is a quadratic expression (e.g., an^2 + bn + c). The difference between consecutive terms is not constant, but the difference between those differences is constant.

Example: 1, 4, 9, 16, ... (n^2).

Key Concept Flip

The second term of a geometric sequence is 6 and the fourth term is 24. Find the possible values of the common ratio.

Answer Flip

Let the first term be 'a' and the common ratio be 'r'. Then ar = 6 and ar^3 = 24. Dividing the second equation by the first, we get r^2 = 4. Therefore, r = 2 or r = -2.

Key Concept Flip

Find the nth term of the sequence: 5, 8, 11, 14, 17, ...

Answer Flip

The common difference (d) = 8 - 5 = 3

For a linear sequence, nth term = dn + (a - d)
where a = first term, d = common difference

nth term = 3n + (5 - 3) = 3n + 2

Check: n=1: 3(1)+2=5 ✓, n=2: 3(2)+2=8 ✓, n=5: 3(5)+2=17 ✓

Key Concept Flip

Find the nth term of the sequence: 3, 12, 27, 48, 75, ...

Answer Flip

Check differences:
First differences: 9, 15, 21, 27 (not constant)
Second differences: 6, 6, 6 (constant → quadratic sequence)

Since second difference = 6, coefficient of n² = 6/2 = 3
So nth term starts with 3n²

Check: 3(1)² = 3 ✓, 3(2)² = 12 ✓, 3(3)² = 27 ✓

nth term = 3n²

For quadratic sequences: if second difference = d, then the n² coefficient = d/2.

Key Concept Flip

What is the next term in the sequence: 2, 6, 18, 54, ...? What type of sequence is this?

Answer Flip

This is a geometric sequence where each term is multiplied by the same number (the common ratio).

Common ratio = 6/2 = 3

Next term = 54 × 3 = 162

General term = a × r^(n-1) = 2 × 3^(n-1)

A geometric sequence has a constant ratio between consecutive terms, unlike an arithmetic sequence which has a constant difference.

Key Concept Flip

The nth term of a sequence is 4n - 7. Find the first three terms and the 20th term.

Answer Flip

Substitute n = 1, 2, 3 and 20:

n = 1: 4(1) - 7 = -3
n = 2: 4(2) - 7 = 1
n = 3: 4(3) - 7 = 5
n = 20: 4(20) - 7 = 73

First three terms: -3, 1, 5
20th term: 73

The common difference is 4 (the coefficient of n).

Key Concept Flip

Write down the first five terms of the Fibonacci sequence and explain the rule.

Answer Flip

First five terms: 1, 1, 2, 3, 5

Rule: each term is the sum of the two previous terms.
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8 (6th term)
5 + 8 = 13 (7th term)

The Fibonacci sequence appears in nature (spiral shells, flower petals, leaf arrangements). It is neither arithmetic nor geometric — it is a recursive sequence defined by its previous two terms.

Key Concept Flip

Find the nth term of the sequence: 1, 4, 9, 16, 25, ...

Answer Flip

These are square numbers: 1², 2², 3², 4², 5², ...

nth term = n²

Check: n=1: 1²=1 ✓, n=4: 4²=16 ✓, n=5: 5²=25 ✓

Recognising common number patterns (square numbers, cube numbers, triangular numbers, powers of 2) helps identify sequences quickly.

Related: 0, 3, 8, 15, 24, ... → these are n² - 1.

Key Concept Flip

How do you find the nth term of a quadratic sequence?

Answer Flip

A quadratic sequence has a constant second difference.

Method:
1. Find first differences (differences between consecutive terms)
2. Find second differences (differences between first differences)
3. If second difference = d, the coefficient of n² is d/2
4. Subtract the n² sequence from the original to get a linear sequence
5. Find the nth term of the linear part
6. Combine: nth term = (d/2)n² + (linear part)

Example: 2, 6, 12, 20 → second diff = 2 → n² term is n². Subtract: 1, 2, 3, 4 → n. So nth term = n² + n.

Key Concept Flip

Is 150 a term in the sequence with nth term 3n + 6? Show your working.

Answer Flip

Set the nth term equal to 150:
3n + 6 = 150
3n = 144
n = 48

Since n = 48 is a positive whole number, yes, 150 is the 48th term of the sequence.

If n had been a fraction or negative, then the number would NOT be a term in the sequence. This method works for any "is X in the sequence?" question.

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2.3 Inequalities 2.5 Quadratics

Key Questions: Sequences