Arithmetic progression 

Arithmetic Progression is a numerical series in which the difference is treated as a constant variable. Because the difference between two successive terms (say 1 and 2) is equal to one, the natural number sequence 1, 2, 3, 4, 5, 6,… is an AP (2 -1). Is is a very easy concept and can be understood with a quick and easy formula. 


Important Points to Remember About Arithmetic Progression

  • An AP is a set of numbers in which each phrase is produced by multiplying the preceding number by a specified integer.
  • The first term is represented by a, the common difference is represented by d, the nth term is represented by an, and the number of terms is represented by n.
  • An AP’s graph is a straight line with the common difference as the slope.
  • The common difference does not necessarily have to be good.

There are three basic forms of progressions in mathematics. They are as follows:

  • Progression in Arithmetic (AP)
  • Progression in Geometry (GP)

Harmonic Progression (HP) A progression is a sort of sequence for which the nth term may be calculated using a formula. The Arithmetic Progression is the most widely used mathematical series, using simple formulas. Let’s have a look at the three sorts of definitions it provides.

  • Definition 1: An AP sequence is a mathematical series in which the difference between two successive terms is always a constant.
  • Definition 2: An arithmetic sequence or progression is a set of numbers in which the second number is generated by adding a fixed number to the first for each pair of successive terms.
  • Definition 3: The common difference of an AP is the set number that must be added to any phrase of an AP to get the following term. Consider the following series: 1, 4, 7, 10, 13, 16,… This is an arithmetic sequence with a common difference of 3.

Differences in Arithmetic Progression are a common occurrence.

The initial term, the common difference between the two terms, and the nth term are employed in this progression for a particular series. If a1, a2, a3,……………., and an is an AP, then the common difference ” d ” may be calculated as follows:

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


When learning about Arithmetic Progression, we come across two important formulae that are connected to:

  • The AP’s nth term
  • The first n terms’ sum

Let’s go through both formulae with some instances.

Example: If there are 15 terms in AP, find the nth term: 1, 2, 3, 4, 5….

Solution: AP: 1, 2, 3, 4, 5,…, and


An = a+(n-1)d, according to the formula we know.

A =1 in the first term

d=2-1 =1 is a common difference.

As a result, a = 1 + (15-1)1 = 1 + 14 = 15

Note: Finite AP refers to the finite fraction of an AP, while arithmetic series refers to the sum of finite AP. The value of a common difference determines how the sequence behaves.

  • If “d” has a positive value, the member terms will increase to positive infinity.
  • If “d” is negative, the member terms expand until they reach negative infinity.

The total number of AP terms is N.


An arithmetic mean is a sequence of terms with a common difference between them that is a constant number. It’s a term that’s used to generalize a group of patterns that we see in our daily lives. Want to learn more then you must enroll in Cuemath classes. Our experts will help you understand the concept from the basics and make you grow in the right direction.