For any sequence a1, a2, a3.... the sequence {a1+a2+a3+ ... +an} is called a series. A series is finite or infinite according as the number of terms added is finite or infinite.


Sequences whose terms follow certain patterns are called progressions.

Arithmetic Progression

An arithmetic progression (AP) is a sequence in which terms increase or decrease regularly by the same constant. A general AP, where a is the first term of AP and d is the common difference of AP, is

a, a+d, a+2d, ..., a+(n-1)d

General Term

Tn = a + (n-1)d

If the all terms of an AP are increased, decreased, multiplied and divided by the same non-zero constant, then they remain in AP.

Three consecutive numbers in AP can be taken as a-d, a, a+d

Four consecutive numbers in AP can be taken as a-3d, a-d, a+d, a+3d

Sum of AP

Sn = n/2[2a+(n-1)d] = n/2[a+l]

Arithmetic Mean

If a, A, b are in AP, then A is called by arithmetic mean.

A = (a+b)/2

Geometric Progression

A sequence is said to be a geometric progression (GP), if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio.

General Term

Tn = arn-1

Sum of Terms

Sn = a(rn-1)/(r-1)

Geometric Mean

The geometric mean (GM) of any two positive numbers a and b is given by √ab. The sequence a, G, b is GP.

G = √ab