Arithemetic Progressions ( A.P. ) !!......

Introduction :-

An Arithemetic Prgression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

e.g:- 1 , 4 , 7 , 10 , 13..........

In this example , first term is 1 and all other terms are obtained by adding a fixed number to the preceding term.

First term is signified by :- a

Common Difference between each term is signified by :- d

Here , a = 1 i.e. first term and d = 3 i.e. common difference.

The fixed number i.e. common difference (d) in A.P. Can be positive , negative or zero.

General form an A.P. :-

An Arithemetic Prgression where a is the first term and d is the common difference is referred to as General form an A.P.

Finite A.P. :-

Some A.P.s have only finte number of terms . Such an A.P. is called as Finite A.P. .

Such Finite A.P.s have the last term (l) .

Infinite A.P. :-

The A.P.s having infinite number of terms is referred to as Infinte A.P.

Such A.P.s doesn't have the last term (l).

nth Term of an A.P. :-

Let a₁ , a₂ , a₃ , …. be an A.P. whose first term a₁ is a and the common difference is d .

Then ,

the second term (a₂) = a + d = a + (2 – 1) d

the third term (a₃) = a₂ + d = (a + d ) + d = a + 2d = a + (3 – 1 ) d

the fourth term (a₄) = a₃ + d = (a + 2d ) + d = a + 3d = a + (4 – 1 ) d

Looking at the pattern above , we can say that the nth term a_n = a + (n – 1 ) d .

So , the nth term a_n of the A.P. with first term a and common difference d is given by:-

a_n = a + ( n – 1 ) d .

an is also called the general term of an A.P. . If there are m terms in the A.P. , then a_m represents the last term which is sometimes donated by l .

Sum of first n terms of an A.P. :-

The sum of first n terms of an A.P. with first term ' a ' and common difference ' d ' is given by :-

S_n = n / 2 [ 2a + (n – 1 ) d ]

It can also be written as :-

S_n = n / 2 [ a + a ( n – 1 ) d ]

i.e. S_n = n / 2 (a + a_n )

So , if there are n terms in the A.P. , then an = l , the last term.

Therefore , the sum can also be given by :-

S_n = n / 2 ( a + l )