Arithmetic Progression

What is Arithmetic Progression?

In mathematics, there are three different types of progressions. They are:

• Arithmetic Progression (AP)
• Geometric Progression (GP)
• Harmonic Progression (HP)

A progression is a special type of sequence for which it is possible to obtain a formula for the nth term.  The Arithmetic Progression is the most commonly used sequence in maths with easy to understand formulas. 

Definition 1: A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.

Definition 2: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.

The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP. Now, let us consider the sequence, 1, 4, 7, 10, 13, 16,…

Notation in Arithmetic Progression

In AP, we will come across some main terms, which are denoted as:

• First term (a)
• Common difference (d)
• nth Term (a_n)
• Sum of the first n terms (S_n)

All three terms represent the property of Arithmetic Progression. We will learn more about these three properties in the next section.

First Term of AP

The AP can also be written in terms of common differences, as follows;

where  “a” is the first term of the progression. 

Common Difference in Arithmetic Progression

In this progression, for a given series, the terms used are the first term, the common difference and nth term. Suppose, a₁, a₂, a₃, ……………., a_n is an AP, then; the common difference “ d ” can be obtained as;

Where “d” is a common difference. It can be positive, negative or zero.

General Form of an AP

Consider an AP to be: a₁, a₂, a₃, ……………., a_n