# Sequences and Series, Arithmetic Progression, General Term of an a. P

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## Arithmetic Progression:

Let us consider the following examples of sequence, of numbers:

Note that in the above four sequences of numbers, the first terms are respectively and .

The first term has an important role in this lesson. Also every following term of the sequence has certain relation with the first term.

What is the relation of the terms with the first term in Example First term , Second term , Third term and Fourth term and so on.

The consecutive terms in the above sequence are obtained by adding 2 to its preceding term.

i.e., the difference between any two consecutive terms is the same.

A finite sequence of numbers with this property is called an arithmetic progression.

A sequence of numbers with finite terms in which the difference between two consecutive terms is the same non-zero number is called the Arithmetic Progression or simply A. P.

The difference between two consecutive terms is called the common difference of the A. P. and is denoted by .

In general, an A. P. whose first term is a and common difference is is written as

Also we use to denote the nth term of the progression.

## General Term of an a. P.:

Let us consider A. P.

Here,

First term

Second term ,

Third term

By observing the above pattern, term can be written as:

Hence, if the first term and the common difference of an A. P. are known then any term of

P. can be determined by the above formula.

Example 1:

The term of an A. P. is and term is , find the term.

Solution:

Let be the first term and be the common difference of the A. P. Then from the formula: , we have

and

We have, ,

Solve equations and to get the values of and .

Subtracting from , we have

Again from ,

Now

Example 2:

Which term of the A. P.: is ?

Solution:

Here

We know that

Therefore, is the 20th term of the given A.P.