Arithmetic progression

Arithmetic progression (also known as an arithmetic sequence) is a sequence of numbers in which the difference between any two successive members is a constant. This difference is referred to as the common difference. Arithmetic progressions are a fundamental concept in number theory, and are used in various areas of mathematics and science.

Definition

An arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant. This difference is called the common difference. If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the nth term of the sequence is given by:

a + (n - 1)d

Properties

Arithmetic progressions have several important properties. These include:

• The nth term of an arithmetic progression can be found using the formula a + (n - 1)d.
• The sum of the first n terms (Sn) of an arithmetic progression can be found using the formula n/2 [2a + (n - 1)d].
• If the common difference is zero, then all the terms of the arithmetic progression are the same.

Examples

Here are some examples of arithmetic progressions:

• The sequence 2, 4, 6, 8, 10, 12, ... is an arithmetic progression with a common difference of 2.
• The sequence 5, 2, -1, -4, -7, ... is an arithmetic progression with a common difference of -3.

Applications

Arithmetic progressions have many applications in various fields, including:

• In mathematics, they are used in the study of number theory, algebra, and geometry.
• In physics, they are used in the study of motion and waves.
• In computer science, they are used in the design of algorithms and data structures.

See also

• Geometric progression
• Harmonic progression
• Fibonacci sequence