Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 21: Sequences & Series

Lesson 3: Geometric Sequences

In this Grade 4 AoPS Introduction to Algebra lesson, students learn how to identify and work with geometric sequences by understanding the common ratio between terms and applying the nth term formula ar^(n-1). The lesson also introduces the geometric mean as the square root of the product of two numbers, connecting it to the structure of geometric sequences. Drawn from Chapter 21 of the AMC 8 and AMC 10 curriculum, practice problems guide students through finding missing terms, solving for unknown common ratios, and modeling real-world exponential growth.

Section 1

Geometric Sequence

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$ , is $r$ , the common ratio. $n$ is greater than or equal to two.

Examples

The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$ .

Section 2

Solving for Common Ratio in Geometric Sequences

Property

To find the common ratio $r$ when given two terms of a geometric sequence, use:

r = \sqrt[n-m]{\frac{a_n}{a_m}}

where

a_n

and

a_m

are the

n

th and

m

th terms respectively, or solve

a_n = a_m \cdot r^{n-m}

for

r

Examples

Section 3

General Term of a Geometric Sequence

Property

The general term of a geometric sequence with first term $a_1$ and the common ratio $r$ is

a_n = a_1 r^{n-1}

Examples

To find the 10th term of a sequence where $a_1 = 4$ and $r = 2$ , we use the formula: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Geometric Sequence

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$ , is $r$ , the common ratio. $n$ is greater than or equal to two.

Examples

The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$ .

Section 2

Solving for Common Ratio in Geometric Sequences

Property

To find the common ratio $r$ when given two terms of a geometric sequence, use:

r = \sqrt[n-m]{\frac{a_n}{a_m}}

where

a_n

and

a_m

are the

n

th and

m

th terms respectively, or solve

a_n = a_m \cdot r^{n-m}

for

r

Examples

Section 3

General Term of a Geometric Sequence

Property

The general term of a geometric sequence with first term $a_1$ and the common ratio $r$ is

a_n = a_1 r^{n-1}

Examples

To find the 10th term of a sequence where $a_1 = 4$ and $r = 2$ , we use the formula: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Geometric Sequence

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$ , is $r$ , the common ratio. $n$ is greater than or equal to two.

Examples

The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$ .

Section 2

Solving for Common Ratio in Geometric Sequences

Property

To find the common ratio $r$ when given two terms of a geometric sequence, use:

r = \sqrt[n-m]{\frac{a_n}{a_m}}

where

a_n

and

a_m

are the

n

th and

m

th terms respectively, or solve

a_n = a_m \cdot r^{n-m}

for

r

Examples

Section 3

General Term of a Geometric Sequence

Property

The general term of a geometric sequence with first term $a_1$ and the common ratio $r$ is

a_n = a_1 r^{n-1}

Examples

To find the 10th term of a sequence where $a_1 = 4$ and $r = 2$ , we use the formula: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .