Recognising Arithmetic and Geometric Progressions — Mathematics

Introduction

Sequences and series appear throughout Pure Mathematics 1 and are a reliable source of marks in the 9709 exam. Before you can find sums, general terms, or solve applied problems, you must be able to recognise whether a given sequence is arithmetic, geometric, or neither. This skill is the essential gateway to all further work on progressions, and examiners regularly test it directly — often as part (a) of a multi-part question.

Core Concept

What is a Sequence?

A sequence is an ordered list of numbers called terms, usually denoted $u_1, u_2, u_3, \ldots$

There are two special families of sequences that the 9709 syllabus focuses on:

Arithmetic Progressions (APs)

An arithmetic progression is a sequence in which each term is obtained from the previous term by adding a fixed constant, called the common difference.

Recognition test: Subtract any term from the next term. If the result is the same constant throughout, the sequence is arithmetic.

u_2 - u_1 = u_3 - u_2 = u_4 - u_3 = \cdots = d \quad \text{(constant)}

The common difference $d$ can be positive, negative, or zero.

Examples of APs:

Sequence	Common difference $d$
$3,\ 7,\ 11,\ 15,\ \ldots$	$d = 4$
$20,\ 15,\ 10,\ 5,\ \ldots$	$d = -5$
$\sqrt{2},\ 2\sqrt{2},\ 3\sqrt{2},\ \ldots$	$d = \sqrt{2}$
$6,\ 6,\ 6,\ 6,\ \ldots$	$d = 0$

Geometric Progressions (GPs)

A geometric progression is a sequence in which each term is obtained from the previous term by multiplying by a fixed constant, called the common ratio.

Recognition test: Divide any term by the previous term. If the result is the same constant throughout, the sequence is geometric.

\frac{u_2}{u_1} = \frac{u_3}{u_2} = \frac{u_4}{u_3} = \cdots = r \quad \text{(constant)}

The common ratio $r$ can be positive, negative, a fraction, or a surd — but it cannot be zero.

Examples of GPs:

Sequence	Common ratio $r$
$2,\ 6,\ 18,\ 54,\ \ldots$	$r = 3$
$80,\ 40,\ 20,\ 10,\ \ldots$	$r = \tfrac{1}{2}$
$5,\ -10,\ 20,\ -40,\ \ldots$	$r = -2$
$1,\ \tfrac{1}{3},\ \tfrac{1}{9},\ \tfrac{1}{27},\ \ldots$	$r = \tfrac{1}{3}$

Sequences That Are Neither

Not every sequence is AP or GP. For instance:

1,\ 4,\ 9,\ 16,\ 25,\ \ldots \quad \text{(square numbers — neither AP nor GP)}

Always perform both tests before concluding.

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