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Arithmetic progression

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An arithmetic progression (AP) is a numerical sequence in which each term, starting from the second one, is equal to the sum of the previous term and a constant number ‘d,’ which is called the common difference of the arithmetic progression.

If the first term of the arithmetic progression is known as ‘a1’ and the common difference is ‘d,’ then any term of the arithmetic progression can be calculated as follows:

a2 = a1 + d

a3 = a2 + d = a1 + 2d

a4 = a3 + d = a1 + 3d

and so on.

To find any term of the progression, the general formula is used:

an = a1 + d(n – 1),


  • an​ is the n-th term of the progression.
  • a1​ is the first term of the progression.
  • n is the number of the term in the progression.
  • d is the common difference between consecutive terms of the progression.

For example, if the first term is a1 = 3, and the common difference is d = 2, then:

a2 = 3 + 2(2 – 1) = 5,

a3 = 3 + 2(3 – 1) = 7,

and so on.

Arithmetic progressions are widely used in mathematics and various fields of science and engineering for modeling growth, motion, cost, and many other processes.