Number Sequences Study Guide

Number sequences are sets of numbers that follow a specific pattern or rule. Understanding number sequences is an important concept in mathematics and can be found in various mathematical problems and real-life applications.

Types of Number Sequences

There are several types of number sequences, including:

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The constant difference is called the "common difference". The general form of an arithmetic sequence is: a, a + d, a + 2d, a + 3d, ...

Formula for the nth term of an arithmetic sequence:

a_n = a₁ + (n - 1)d

Geometric Sequences

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio". The general form of a geometric sequence is: a, ar, ar², ar³, ...

Formula for the nth term of a geometric sequence:

a_n = a₁ * r^(n-1)

Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence starts as: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Square Number Sequence

The square number sequence is a sequence of numbers that are the squares of natural numbers. The sequence starts as: 1, 4, 9, 16, 25, 36, ...

Cubic Number Sequence

The cubic number sequence is a sequence of numbers that are the cubes of natural numbers. The sequence starts as: 1, 8, 27, 64, 125, ...

Study Tips for Number Sequences

Practice identifying patterns in number sequences.
Understand the difference between arithmetic and geometric sequences.
Memorize the formulas for the nth term of arithmetic and geometric sequences.
Solve a variety of sequence problems to reinforce your understanding.
Use visual aids and diagrams to help comprehend the sequences.

By understanding and practicing number sequences, you'll be better prepared to solve problems related to sequences and series in mathematics.

◂Math Worksheets and Study Guides Eighth Grade. Similarity and scale

Using Similar Polygons

Worksheet/Answer key

Similar Polygons

Worksheet/Answer key

Using Similar Polygons

Worksheet/Answer key

Similar Polygons

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

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

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

Number Sequences Study Guide

Types of Number Sequences

Arithmetic Sequences

Formula for the nth term of an arithmetic sequence:

Geometric Sequences

Formula for the nth term of a geometric sequence:

Fibonacci Sequence

Square Number Sequence

Cubic Number Sequence

Study Tips for Number Sequences

[Number Sequences] Related Worksheets and Study Guides:

◂Math Worksheets and Study Guides Eighth Grade. Similarity and scale

The resources above cover the following skills: