**Sequence**: A set of numbers arranged in some definite order and formed according to some rules is called a sequence.

**Progression**: The sequence that follows a certain pattern is called progression.

An **arithmetic progression** is a sequence of numbers such that the difference of any two successive members is a constant.

** For example**, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference

** Second example:** the sequence 3, 5, 7, 9, 11,... is an arithmetic progression

with common difference

with common difference

**Notation**

We denote by **d** the common difference.

By **a _{n}** we denote the

By **S _{n}** we denote the sum of the first n elements of an arithmetic series.

**Properties**

**a _{1} + a_{n} = a_{2} + a_{n-1} = ... = a_{k} + a_{n-k+1}**

and

**a _{n} = ½(a_{n-1} + a_{n+1})**

Sample: let 1, 11, 21, 31, 41, 51... be an arithmetic progression.

51 + 1 = 41 + 11 = 31 + 21

and

11 = (21 + 1)/2

21 = (31 + 11)/2...

If the initial term of an arithmetic progression is ** a_{1}** and the common difference of successive members is

**a _{n} = a_{1} + (n - 1)d, n = 1, 2, ...**

The sum ** S** of the first

**S = ½(a _{1} + l)n**

where ** a_{1}** is the first term and

or

**S = ½(2a _{1} + d(n-1))n**

(i) If is **a _{1 }**given, then d = an – an-1 common difference

(ii) If is given, then term is given by

(iii) If a, b, c are in A.P., then 2b = a + c.

(iv) If a sequence has n terms, its term from the end term from the

beginning.

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