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

Lesson 1: Arithmetic Sequences

In this Grade 4 AoPS Introduction to Algebra lesson, students learn to identify arithmetic sequences, define the common difference, and apply the nth term formula a + (n−1)d to find any term in a sequence. Through problems involving negative terms, missing terms, and averages, students practice deriving formulas from first principles rather than memorization. This lesson is part of Chapter 21 on Sequences and Series, building foundational skills for AMC 8 and AMC 10 competition problem solving.

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

Finding Terms by Counting Steps

Property

To find any term in an arithmetic sequence, count the number of steps from the first term and multiply by the common difference: $a_n = a_1 + (\text{number of steps}) \times d$

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

Finding Terms by Counting Steps

Property

To find any term in an arithmetic sequence, count the number of steps from the first term and multiply by the common difference: $a_n = a_1 + (\text{number of steps}) \times d$

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

Finding Terms by Counting Steps

Property

To find any term in an arithmetic sequence, count the number of steps from the first term and multiply by the common difference: $a_n = a_1 + (\text{number of steps}) \times d$

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: $a_{20} = 5 + (20 - 1)4 = 5 + 19 \cdot 4 = 81$ .