Series

In mathematics, a series is the sum of the terms of a sequence. A series can be finite or infinite. The sum of a finite series is called a finite sum, while the sum of an infinite series is called an infinite sum.

Types of Series

There are several types of series, including arithmetic series, geometric series, and harmonic series. Each type of series has its own distinct pattern and method for finding the sum of its terms.

Arithmetic Series

An arithmetic series is a series in which each term is obtained by adding a constant difference to the previous term. The sum of the terms of an arithmetic series can be found using the formula:

S_n = n/2 * (a₁ + a_n)

Where S_n is the sum of the first n terms, n is the number of terms, a₁ is the first term, and a_n is the nth term.

Geometric Series

A geometric series is a series in which each term is obtained by multiplying the previous term by a constant ratio. The sum of the terms of a geometric series can be found using the formula:

S_n = a₁ * (1 - rⁿ) / (1 - r)

Where S_n is the sum of the first n terms, a₁ is the first term, r is the common ratio, and n is the number of terms.

Harmonic Series

A harmonic series is the sum of the reciprocals of the terms of a sequence. The sum of the terms of a harmonic series can be expressed as:

S_n = 1 + 1/2 + 1/3 + ... + 1/n

Convergence and Divergence

When dealing with infinite series, it is important to understand whether a series converges (has a finite sum) or diverges (has no finite sum). Tests such as the divergence test, comparison test, ratio test, and root test can be used to determine the convergence or divergence of a series.

Study Guide

Identify the type of series (arithmetic, geometric, harmonic).
Determine the common difference or ratio for the series.
Find the sum of the series using the appropriate formula.
For infinite series, use convergence tests to determine convergence or divergence.

Practice solving problems involving different types of series and use convergence tests to determine the behavior of infinite series. Understanding the properties and behavior of series is crucial for various areas of mathematics and its applications in real-life scenarios.

◂Math Worksheets and Study Guides Sixth Grade. Formulas

Study Guide

Formulas

Activity Lesson

Equivalent Expressions (Distributive Property)

The resources above cover the following skills:

Algebra (NCTM)

Use mathematical models to represent and understand quantitative relationships.

Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

Grade 6 Curriculum Focal Points (NCTM)

Algebra: Writing, interpreting, and using mathematical expressions and equations

Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation. They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table.

Connections to the Grade 6 Focal Points (NCTM)

Measurement and Geometry: Problems that involve areas and volumes, calling on students to find areas or volumes from lengths or to find lengths from volumes or areas and lengths, are especially appropriate. These problems extend the students' work in grade 5 on area and volume and provide a context for applying new work with equations.