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

Lesson 4: Geometric Series

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to find the sum of a geometric series using the formula a(r^n − 1)/(r − 1), and explore the special case where r equals 1. The lesson also introduces infinite geometric series, including the convergence formula a/(1 − r) for |r| < 1, and the concepts of convergent, divergent, and indeterminate series. Students apply these ideas through structured problem-solving that builds the general summation formula from first principles.

Section 1

Multiply-and-Subtract Method for Geometric Series

Property

To derive the geometric series formula, multiply the series by the common ratio $r$ and subtract from the original series to eliminate most terms: If $S = a + ar + ar^2 + \cdots + ar^{n-1}$ , then $rS = ar + ar^2 + ar^3 + \cdots + ar^n$ , so $S - rS = a - ar^n$ .

Examples

Section 2

Geometric Series with Common Ratio r = 1

Property

When the common ratio $r = 1$ in a geometric series, all terms are equal to the first term $a$ . The sum of $n$ terms is:

S_n = na

Section 3

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Multiply-and-Subtract Method for Geometric Series

Property

Examples

Section 2

Geometric Series with Common Ratio r = 1

Property

When the common ratio $r = 1$ in a geometric series, all terms are equal to the first term $a$ . The sum of $n$ terms is:

S_n = na

Section 3

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.

Overview

Lesson overview

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

Overview

Section 1

Multiply-and-Subtract Method for Geometric Series

Property

Examples

Section 2

Geometric Series with Common Ratio r = 1

Property

When the common ratio $r = 1$ in a geometric series, all terms are equal to the first term $a$ . The sum of $n$ terms is:

S_n = na

Section 3

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.