Learn on PengienVision, Algebra 2Chapter 1: Linear Functions and Systems

Lesson 4: Arithmetic Sequences and Series

In this Grade 11 enVision Algebra 2 lesson, students learn to identify arithmetic sequences by finding the common difference and write both recursive and explicit definitions to represent any term. The lesson also introduces arithmetic series and sigma notation, connecting linear function concepts to real-world applications like auditorium seating arrangements.

Section 1

Arithmetic Sequence

Property

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, $a_n - a_{n-1}$ , is $d$ , the common difference, for $n$ greater than or equal to two.

Examples

The sequence $6, 11, 16, 21, 26, \ldots$ is arithmetic because the difference is always 5. The common difference is $d=5$ .

Section 2

Recursive Definition of Arithmetic Sequences

Property

The recursive definition of an arithmetic sequence is written as:

a_n = \begin{cases} a_1, & \text{if } n = 1 \\ a_{n-1} + d, & \text{if } n > 1 \end{cases}

where $a_1$ is the first term and $d$ is the common difference.

Examples

Section 3

General Term of an Arithmetic Sequence

Property

The general term of an arithmetic sequence with first term $a_1$ and the common difference $d$ is

a_n = a_1 + (n - 1)d

Examples

To find the 20th term of a sequence where $a_1 = 5$ and $d = 4$ , use the formula: $a_{20} = 5 + (20 - 1)4 = 5 + 19 \cdot 4 = 81$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Arithmetic Sequence

Property

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, $a_n - a_{n-1}$ , is $d$ , the common difference, for $n$ greater than or equal to two.

Examples

The sequence $6, 11, 16, 21, 26, \ldots$ is arithmetic because the difference is always 5. The common difference is $d=5$ .

Section 2

Recursive Definition of Arithmetic Sequences

Property

The recursive definition of an arithmetic sequence is written as:

a_n = \begin{cases} a_1, & \text{if } n = 1 \\ a_{n-1} + d, & \text{if } n > 1 \end{cases}

where $a_1$ is the first term and $d$ is the common difference.

Examples

Section 3

General Term of an Arithmetic Sequence

Property

The general term of an arithmetic sequence with first term $a_1$ and the common difference $d$ is

a_n = a_1 + (n - 1)d

Examples

To find the 20th term of a sequence where $a_1 = 5$ and $d = 4$ , use the formula: $a_{20} = 5 + (20 - 1)4 = 5 + 19 \cdot 4 = 81$ .

Overview

Lesson overview

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

Overview

Section 1

Arithmetic Sequence

Property

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, $a_n - a_{n-1}$ , is $d$ , the common difference, for $n$ greater than or equal to two.

Examples

The sequence $6, 11, 16, 21, 26, \ldots$ is arithmetic because the difference is always 5. The common difference is $d=5$ .

Section 2

Recursive Definition of Arithmetic Sequences

Property

The recursive definition of an arithmetic sequence is written as:

a_n = \begin{cases} a_1, & \text{if } n = 1 \\ a_{n-1} + d, & \text{if } n > 1 \end{cases}

where $a_1$ is the first term and $d$ is the common difference.

Examples

Section 3

General Term of an Arithmetic Sequence

Property

The general term of an arithmetic sequence with first term $a_1$ and the common difference $d$ is

a_n = a_1 + (n - 1)d

Examples

To find the 20th term of a sequence where $a_1 = 5$ and $d = 4$ , use the formula: $a_{20} = 5 + (20 - 1)4 = 5 + 19 \cdot 4 = 81$ .