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.

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