Learn on PengiBig Ideas Math, Algebra 1Chapter 6: Exponential Functions and Sequences

Lesson 6: Geometric Sequences

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

Section 1

Geometric Sequence

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.

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

General Term of a Geometric Sequence

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}

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

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Section 1

Geometric Sequence

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.

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

General Term of a Geometric Sequence

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}

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

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

Overview

Section 1

Geometric Sequence

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.

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

General Term of a Geometric Sequence

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}

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