Sequence and Series

Arithmetic Progression (AP)

  • An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant ‘d’ to the preceding term. The constant d is called the common difference.


  • An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), …

where a = the first term, d = the common difference


  • If a, b, c are in AP then b = (a + c)/2
  • nth term of an arithmetic progression

tn = a + (n – 1)d

where tn = nth term, a= the first term, d= common difference


  • Number of terms of an arithmetic progression


where n = number of terms, a= the first term , l = last term, d= common difference



no of term
sum of first n terms
the sum of first n terms


To solve most of the problems related to AP, the terms can be conveniently taken as

3 terms: (a – d), a, (a +d)

4 terms: (a – 3d), (a – d), (a + d), (a +3d)

5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)

Harmonic Progression(HP)

harmonic progression
harmonic progression

Geometric Progression (GP)

Geometric Progression (GP) is a sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.

geometric progression

sum of GP
sum of infinite
geometric progression
power series