For any sequence a_{1}, a_{2}, a_{3}.... the sequence {a_{1}+a_{2}+a_{3}+ ... +a_{n}} is called a series. A series is finite or infinite according as the number of terms added is finite or infinite.

**Progressions**

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**

T_{n} = 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**

T_{n} = ar^{n-1}

**Sum of Terms**

S_{n} = a(r^{n}-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