Learn on PengiOpenstax Intermediate Algebra 2EChapter 12: Sequences, Series and Binomial Theorem

Lesson 12.3: Geometric Sequences and Series

In this lesson from OpenStax Intermediate Algebra 2E, students learn to identify geometric sequences by finding the common ratio between consecutive terms, derive the general nth term formula, and calculate finite and infinite geometric series sums. The lesson also covers real-world applications of geometric sequences and series. It is part of Chapter 12 and is suited for intermediate algebra students at the high school or early college level.

Section 1

📘 Geometric Sequences and Series

New Concept

Today, we'll explore geometric sequences, where each term is found by multiplying the previous one by a constant ratio. We'll learn to find any term, sum up finite and infinite series, and see how these concepts apply to finance.

What’s next

Next, you’ll master identifying these sequences through worked examples, then tackle practice problems to find terms and calculate sums of series.

Section 2

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 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: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .

Section 4

Sum of a Finite Geometric Sequence

Property

The sum, $S_n$ , of the first $n$ terms of a geometric sequence is

S_n = \frac{a_1 (1 - r^n)}{1 - r}

where $a_1$ is the first term and $r$ is the common ratio, and $r$ is not equal to one.

Section 5

Sum of an Infinite Geometric Series

Property

For an infinite geometric series whose first term is $a_1$ and common ratio $r$ ,

If $|r| < 1$ , the sum is

S = \frac{a_1}{1 - r}

Section 6

Value of an Annuity

Property

For a principal, $P$ , invested at the end of a compounding period, with an interest rate, $r$ , which is compounded $n$ times a year, the new balance, $A_t$ , after $t$ years, is

A_t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}

Examples

If parents invest 150 dollars per month for their child at 5% annual interest compounded monthly, after 18 years the account will have $A_{18} = \frac{150 \left( \left(1 + \frac{0.05}{12}\right)^{12 \cdot 18} - 1 \right)}{\frac{0.05}{12}} \approx 43,650.25$ dollars.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 12: Sequences, Series and Binomial Theorem

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 12.1: Sequences
Learn Practice
Lesson 2
Lesson 12.2: Arithmetic Sequences
Learn Practice
Lesson 3Current
Lesson 12.3: Geometric Sequences and Series
Learn Practice
Lesson 4
Lesson 12.4: Binomial Theorem
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Geometric Sequences and Series

New Concept

What’s next

Next, you’ll master identifying these sequences through worked examples, then tackle practice problems to find terms and calculate sums of series.

Section 2

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 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: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .

Section 4

Sum of a Finite Geometric Sequence

Property

The sum, $S_n$ , of the first $n$ terms of a geometric sequence is

S_n = \frac{a_1 (1 - r^n)}{1 - r}

where $a_1$ is the first term and $r$ is the common ratio, and $r$ is not equal to one.

Section 5

Sum of an Infinite Geometric Series

Property

For an infinite geometric series whose first term is $a_1$ and common ratio $r$ ,

If $|r| < 1$ , the sum is

S = \frac{a_1}{1 - r}

Section 6

Value of an Annuity

Property

For a principal, $P$ , invested at the end of a compounding period, with an interest rate, $r$ , which is compounded $n$ times a year, the new balance, $A_t$ , after $t$ years, is

A_t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}

Examples

If parents invest 150 dollars per month for their child at 5% annual interest compounded monthly, after 18 years the account will have $A_{18} = \frac{150 \left( \left(1 + \frac{0.05}{12}\right)^{12 \cdot 18} - 1 \right)}{\frac{0.05}{12}} \approx 43,650.25$ dollars.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 12: Sequences, Series and Binomial Theorem

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 12.1: Sequences
Learn Practice
Lesson 2
Lesson 12.2: Arithmetic Sequences
Learn Practice
Lesson 3Current
Lesson 12.3: Geometric Sequences and Series
Learn Practice
Lesson 4
Lesson 12.4: Binomial Theorem
Learn Practice

Overview

Lesson overview

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

Overview

Section 1

📘 Geometric Sequences and Series

New Concept

What’s next

Next, you’ll master identifying these sequences through worked examples, then tackle practice problems to find terms and calculate sums of series.

Section 2

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 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: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$ .

Section 4

Sum of a Finite Geometric Sequence

Property

The sum, $S_n$ , of the first $n$ terms of a geometric sequence is

S_n = \frac{a_1 (1 - r^n)}{1 - r}

where $a_1$ is the first term and $r$ is the common ratio, and $r$ is not equal to one.

Section 5

Sum of an Infinite Geometric Series

Property

For an infinite geometric series whose first term is $a_1$ and common ratio $r$ ,

If $|r| < 1$ , the sum is

S = \frac{a_1}{1 - r}

Section 6

Value of an Annuity

Property

For a principal, $P$ , invested at the end of a compounding period, with an interest rate, $r$ , which is compounded $n$ times a year, the new balance, $A_t$ , after $t$ years, is

A_t = \frac{P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}

Examples

If parents invest 150 dollars per month for their child at 5% annual interest compounded monthly, after 18 years the account will have $A_{18} = \frac{150 \left( \left(1 + \frac{0.05}{12}\right)^{12 \cdot 18} - 1 \right)}{\frac{0.05}{12}} \approx 43,650.25$ dollars.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 12: Sequences, Series and Binomial Theorem

Back to Openstax Intermediate Algebra 2E

Lesson 1
Lesson 12.1: Sequences
Learn Practice
Lesson 2
Lesson 12.2: Arithmetic Sequences
Learn Practice
Lesson 3Current
Lesson 12.3: Geometric Sequences and Series
Learn Practice
Lesson 4
Lesson 12.4: Binomial Theorem
Learn Practice

Lesson 12.3: Geometric Sequences and Series

New Concept

What’s next

Property

Examples

Property

Examples

Property

Property

Property

Examples

Chapter 12: Sequences, Series and Binomial Theorem

Lesson 12.1: Sequences

Lesson 12.2: Arithmetic Sequences

Lesson 12.3: Geometric Sequences and Series

Lesson 12.4: Binomial Theorem

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Property

Property

Examples

Chapter 12: Sequences, Series and Binomial Theorem

Lesson 12.1: Sequences

Lesson 12.2: Arithmetic Sequences

Lesson 12.3: Geometric Sequences and Series

Lesson 12.4: Binomial Theorem

Lesson 12.1: Sequences

Lesson 12.2: Arithmetic Sequences

Lesson 12.3: Geometric Sequences and Series

Lesson 12.4: Binomial Theorem