Definition: A sequence (progression) is a set of numbers in a definite order with a definite rule of obtaining the numbers.

__Arithmetic Progression (A. P.)__

Definition: An A.P. is a sequence whose terms increase or decrease by a fixed number , called the common difference of the A.P.

__n__^{th} Term

^{th}Term

If a is the first term and d the common difference, the A.P. can be written as a , a + d , a + 2d , ……

The n^{th} term a_{n} is given by a_{n} = a + (n − 1)d

### Sum of n Terms

The sum S_{n} of the first n terms of such an A.P. is given by

S_{n} = ( n/2) [2a + (n − 1)d ]

S_{n} = ( n/2) [ a + l ]

where l is the last term (i.e. the nth term of the A.P.).

__Notes:__

* If a fixed number is added (subtracted) to each term of a given A.P. then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.

* If each term of an A.P. is multiplied by a fixed number (say k) (or divided by a non-zero fixed number), the resulting sequence is also an A.P. with the common difference multiplied by k.

* If a_{1}, a_{2}, a_{3}…..and b_{1}, b_{2}, b_{3}…are two A.P.’s with common differences d and d’ respectively then a_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3} , …is also an A.P. with common difference d + d’

* If we have to take three terms in an A.P., it is convenient to take them as a − d, a, a + d.

In general , we take a − rd , a − (r − 1)d ,……a − d , a , a + d ,…….a + rd in case we have to take (2r + 1) terms in an A.P

* If we have to take four terms , we take a − 3d , a − d , a + d , a + 3d.

In general, we take a − (2r − 1)d, a − (2r − 3)d,….a − d , a + d,…..a + (2r − 1)d, in case we have to take 2r terms in an A.P.

* If a_{1}, a_{2}, a_{3}, ….. a_{n} are in A.P. then a_{1} + a_{n} = a_{2} + a_{n−1} = a_{3} + a_{n−2} = . . . . . and so on.

* If n^{th} term of any sequence is a linear expression in n, then the sequence is an AP, whose common difference is the coefficient of n.

* If sum of n terms of any sequence is a quadratic in n, whose constant term is zero, then the sequence is an AP, whose common difference is twice the coefficient of n^{2}. If the constant term is non-zero, then it is an A.P. from second term onwards.

* If {tn} is an A.P., then the common difference, d, is given by d = (t_{p}−t_{q})/(p−q) , (p , q∈N)

__Arithmetic Mean(s):__

* If three terms are in A.P., then the middle term is called the arithmetic mean (A.M.) between the other two i.e. if a, b, c are in A.P. then b = (a + c)/2 is the A.M. of a and c.

* If a_{1}, a_{2}, … a_{n} are n numbers then the arithmetic mean (A) of these numbers is

* The n numbers A_{1}, A_{2}…..A_{n} are said to be A.M.’ s between the numbers a and b if a, A_{1}, A_{2} ,…..A_{n}, b are in A.P. If d is the common difference of this A.P. then b = a + (n + 2 − 1)d

where A_{r} is the r^{th} mean