Question

What is Arithmetic Progression?​

Answer 1

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An Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

For example, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.

Answer 2

Answer

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.

First Term of AP

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

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

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

Also, check:

Geometric Progression Sum Of Gp

Arithmetic Progression For Class 10

Important Questions Class 10 Maths Chapter 5 Arithmetic Progressions

General Form of an A. P

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

Position of Terms Representation of Terms Values of Term

1 a1 a = a + (1-1) d

2 a2 a + d = a + (2-1) d

3 a3 a + 2d = a + (3-1) d

4 a4 a + 3d = a + (4-1) d

. . .

. . .

. . .

. . .

n an a + (n-1)d

Formulas

There are two major formulas we come across when we learn about Arithmetic Progression, which is related to:

The nth term of AP

Sum of the first n terms

Let us learn here both the formulas with examples.

nth Term of an AP

The formula for finding the n-th term of an AP is: