Learn on PengienVision, Algebra 1Chapter 6: Exponents and Exponential Functions

Lesson 4: Geometric Sequences

In this Grade 11 enVision Algebra 1 lesson, students learn to identify geometric sequences by finding a common ratio between consecutive terms and distinguish them from arithmetic sequences. Students practice writing both recursive formulas and explicit formulas to find any term in a geometric sequence, then explore how geometric sequences connect to exponential functions. Real-world contexts, such as blog subscriber growth and festival attendance decline, illustrate how a constant ratio models exponential change.

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

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 3

Writing the Explicit Formula for a Geometric Sequence

Property

The explicit formula for a geometric sequence is used to find any term in the sequence directly. The formula is given by:

a_n = a_1 \cdot r^{n-1}

where $a_n$ is the $n$ -th term, $a_1$ is the first term, and $r$ is the common ratio.

Examples

To write the explicit formula for a sequence with a first term of $3$ and a common ratio of $2$ , we substitute $a_1 = 3$ and $r = 2$ into the formula: $a_n = 3 \cdot 2^{n-1}$ .
For the sequence $5, 15, 45, \dots$ , the first term is $a_1 = 5$ . The common ratio is $r = \frac{15}{5} = 3$ . The explicit formula is $a_n = 5 \cdot 3^{n-1}$ .
For the sequence $100, 50, 25, \dots$ , the first term is $a_1 = 100$ and the common ratio is $r = \frac{50}{100} = \frac{1}{2}$ . The explicit formula is $a_n = 100 \cdot (\frac{1}{2})^{n-1}$ .

Explanation

The explicit formula, also known as the general term, allows you to calculate any term in a geometric sequence without having to find all the preceding terms. It is defined using the sequence''s first term ( $a_1$ ) and its common ratio ( $r$ ). To write the formula, you simply substitute the known values of $a_1$ and $r$ into the standard equation $a_n = a_1 \cdot r^{n-1}$ . This formula is a type of exponential function, where the term number $n$ acts as the variable in the exponent.

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

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 3

Writing the Explicit Formula for a Geometric Sequence

Property

The explicit formula for a geometric sequence is used to find any term in the sequence directly. The formula is given by:

a_n = a_1 \cdot r^{n-1}

where $a_n$ is the $n$ -th term, $a_1$ is the first term, and $r$ is the common ratio.

Examples

To write the explicit formula for a sequence with a first term of $3$ and a common ratio of $2$ , we substitute $a_1 = 3$ and $r = 2$ into the formula: $a_n = 3 \cdot 2^{n-1}$ .
For the sequence $5, 15, 45, \dots$ , the first term is $a_1 = 5$ . The common ratio is $r = \frac{15}{5} = 3$ . The explicit formula is $a_n = 5 \cdot 3^{n-1}$ .
For the sequence $100, 50, 25, \dots$ , the first term is $a_1 = 100$ and the common ratio is $r = \frac{50}{100} = \frac{1}{2}$ . The explicit formula is $a_n = 100 \cdot (\frac{1}{2})^{n-1}$ .

Explanation

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

Recursive Formula for Geometric Sequences (Property Form)

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

a_n = r \cdot a_{n-1}

where

r

is the common ratio and

a_{n-1}

is the previous term.

Examples

Section 3

Writing the Explicit Formula for a Geometric Sequence

Property

The explicit formula for a geometric sequence is used to find any term in the sequence directly. The formula is given by:

a_n = a_1 \cdot r^{n-1}

where $a_n$ is the $n$ -th term, $a_1$ is the first term, and $r$ is the common ratio.

Examples

To write the explicit formula for a sequence with a first term of $3$ and a common ratio of $2$ , we substitute $a_1 = 3$ and $r = 2$ into the formula: $a_n = 3 \cdot 2^{n-1}$ .
For the sequence $5, 15, 45, \dots$ , the first term is $a_1 = 5$ . The common ratio is $r = \frac{15}{5} = 3$ . The explicit formula is $a_n = 5 \cdot 3^{n-1}$ .
For the sequence $100, 50, 25, \dots$ , the first term is $a_1 = 100$ and the common ratio is $r = \frac{50}{100} = \frac{1}{2}$ . The explicit formula is $a_n = 100 \cdot (\frac{1}{2})^{n-1}$ .