■Arithmetical  Progression (A.P.)  :   If  each term of  a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is known as common difference of  the A.P.

An A.P. with first term = a & common difference = d is given by :

a, (a + d), (a + 2d), (a + 3d), …

        The n-th term of  the A.P. with 1^st  term = a & common difference = d is given by :

T_n = a + (n – 1)d.

        The sum of  the n terms of  an A.P. is given by :

S_n =  [2a  +  (n – 1) d]  

     =  ( first term + last term)  =  ( a  +  l  )     :  Where,   l   =  last term

e.g.  (i) How many numbers between 16 and 200 are divisible by 7 ?

 Soln. The required numbers are 21, 28, 35, …, 196.

This is clearly and A.P. in which a = 21 & d = 7.

Let the number of  terms of  this A.P. be n.

Then, T_n = 196  a + (n – 1)d = 196,    i.e. 21 + (n – 1)  7 = 196

n = 26.

Required number  of  numbers  =  26.

Note : Another method to solve the above example is  :

For the given sequence,  = 7n + 14

 7n + 14 = 196

n = 26

e.g. (ii) Find the sum of  first  50  natural numbers.

 Soln. We have to find : 1 + 2 + 3 +…+ 50

This is an A.P. in which a = 1, d = 1 and n = 50.

Required Sum = (first term + last term).

=  (1 + 50)

= 1275.

Note : Another method to solve the above example  :

We know that,     =    =  1  +  2  +  3  + . . . +  n   =

   Here,  1 + 2 + 3 +…+ 50   =      =   1275