INTRODUCTION:
An Arithmetic Progression is a sequence of numbers in which we get each term by adding a particular number to the previous term, except the first term.
- Each number in the sequence is known as term.
- The fixed difference between each term with its preceding term is known as common difference. It can be positive, negative or zero. It is represented as ‘d’.
Some Examples of Arithmetic Progressions:
General form of Arithmetic Progression:
a, a + d, a + 2d, a + 3d,
where the first term is ‘a’ and the common difference is ‘d’.
Example: Given sequence is 2, 5, 8, 11, 14,…
Here, a = 2 and d = 3
d = 5 – 2 = 8 – 5 = 11 – 8 = 3
First term is a = 2
Second term is a + d = 2 + 3 = 5
Third term is a + 2d = 2 + 6 = 8 and so on.
If an is the nth term,a1 is the first term, n is the number of terms in the sequence and d is a common difference then the nth term of an Arithmetic Progression will be an = a + (n – 1)d.
A sequence containing a definite number of terms is called a finite sequence.
A sequence containing an indefinite number of terms is called a infinite sequence.
The sequences in which each term other than the first or last is related to it’s suceeding term having a fixed rule is called a Geometric Progression.
An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.
The fixed number d is called the common difference.
The general form of an AP is a , a + d, a + 2d, a + 3d,. . .
A given list of numbers a_{1}, a_{2} , a_{3}… … … … is an AP, if the differences a_{1} - a_{2} , a_{3} - a_{2} , a_{4} - a_{3} …………… give the same value, i.e.., if a_{k + 1} - a_{k} is the same for different values of k.
In an AP with the first term a and common difference d, the nth term ( or the general term) is given by a_{n} = a + (n - 1) d.
The sum of the first n terms of an AP is given by : S_{n} = \frac{n}{2} [ 2a + n - 1 ] d