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

Lesson 2: Arithmetic Series

In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students learn how to find the sum of an arithmetic series using the formula: number of terms multiplied by the average of the first and last terms, expressed as n[2a + (n−1)d]/2. The lesson walks through Gauss's classic method of pairing terms, and covers special cases including the sum of the first n positive integers (n(n+1)/2) and the sum of the first n odd integers (n²). Students also practice setting up algebraic expressions to solve multi-step arithmetic series problems from the AMC 8 and AMC 10.

Section 1

Pairing Terms Strategy for Arithmetic Series

Property

In an arithmetic series, terms equidistant from the ends have the same sum: $(a_1 + a_n) = (a_2 + a_{n-1}) = (a_3 + a_{n-2}) = \ldots$

This allows pairing terms to find the sum efficiently.

Section 2

Sum of an Arithmetic Sequence

Property

The sum, $S_n$ , of the first $n$ terms of an arithmetic sequence is

S_n = \frac{n}{2}(a_1 + a_n)

where $a_1$ is the first term and $a_n$ is the $n$ th term.

Section 3

Alternative Formula for Arithmetic Series Sum

Property

The sum of an arithmetic series can be calculated using the alternative formula:

S_n = \frac{n[2a + (n-1)d]}{2}

where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Pairing Terms Strategy for Arithmetic Series

Property

In an arithmetic series, terms equidistant from the ends have the same sum: $(a_1 + a_n) = (a_2 + a_{n-1}) = (a_3 + a_{n-2}) = \ldots$

This allows pairing terms to find the sum efficiently.

Section 2

Sum of an Arithmetic Sequence

Property

The sum, $S_n$ , of the first $n$ terms of an arithmetic sequence is

S_n = \frac{n}{2}(a_1 + a_n)

where $a_1$ is the first term and $a_n$ is the $n$ th term.

Section 3

Alternative Formula for Arithmetic Series Sum

Property

The sum of an arithmetic series can be calculated using the alternative formula:

S_n = \frac{n[2a + (n-1)d]}{2}

where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference.

Overview

Lesson overview

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

Overview

Section 1

Pairing Terms Strategy for Arithmetic Series

Property

In an arithmetic series, terms equidistant from the ends have the same sum: $(a_1 + a_n) = (a_2 + a_{n-1}) = (a_3 + a_{n-2}) = \ldots$

This allows pairing terms to find the sum efficiently.

Section 2

Sum of an Arithmetic Sequence

Property

The sum, $S_n$ , of the first $n$ terms of an arithmetic sequence is

S_n = \frac{n}{2}(a_1 + a_n)

where $a_1$ is the first term and $a_n$ is the $n$ th term.

Section 3

Alternative Formula for Arithmetic Series Sum

Property

The sum of an arithmetic series can be calculated using the alternative formula:

S_n = \frac{n[2a + (n-1)d]}{2}

where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference.