Arithmetic Progressions

Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an AP, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.

What you’ll learn

- AP(SEQUENCE AND SERIES).

Course Content

- AP –> 6 lectures • 39min.

Requirements

- No pre course required.

Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an AP, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.

If we observe in our regular lives, we come across Arithmetic progression quite often. For example, Roll numbers of students in a class, days in a week or months in a year. This pattern of series and sequences has been generalized in Maths as progressions.

- Definition
- Common Difference
- First Term

- General Form
- Nth Term
- Sum of Nth Term
- Formula List
- Questions and Solutions
- Problems to Solve

Definition

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. Let’s have a look at its three different types of definitions.

**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.

**Definition 3:** 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,… is considered as an arithmetic sequence with common difference 3.

Notation in AP

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

- Common difference (d)
- nth Term (an)
- Sum of the first n terms (Sn)

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

Common Difference in Arithmetic Progression

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

d = a2 – a1 = a3 – a2 = ……. = an – an – 1

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