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