arithmetic progression

Arithmetic progression (A.P)

An  Arithmetic  progression  is  defined  as  that  in  which the difference between any two terms is always constant (fixed).

                     Or

Any quantity is increase or decrease by the same constant then such quantities form a series which is known as an Arithmetic Progression (A.P).

This constant is known as a  common difference of the Arithmetic Progression (A.P).

So , we can say that Arithmetic Progression (A.P) is increase or decrease by a fixed number ( common difference) .

nth term of an Arithmetic Progression (A.P).

                 Let

a = 1st term

d = common difference

Formation of an Arithmetic Progression(A.P).

1st term =  a

2nd term =  a + d

3rd term =  a + 2d

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nth term =  a + (n-1)d

Tn = a + (n-1)d

nth term from last :

Let l be the last term of an arithmetic progression (A.P)

Tn = a + (n-1)d = l

Common difference :

          d = Tn – Tn-1

 = a + (n-1)d  – [a + (n-2)d]  

= -d

Then , nth term from last

          Tn =  l – (n-1)d

Sum of First n terms of an A.P

Let Sn be the summation of first n terms of an A.P

Sn =  a +(a+d) + (a+2d) +

     ( a+3d)+…+ (a + (n-1)d ).

This can be also written as

Sn = (a + (n-1)d )+(a+(n-2)d)

            +…  + (a+d) +  a.

Add both equation—

2Sn =2a + (n-1)d + 2a + (n-1)d +

2a + (n-1)d + …… n terms

Sn = n/2[2a + (n-1)d]

Note :

        Tn =  Sn – Sn-1

Important points :

  1. If  a1 , a2 , a3 , ……  are in A.P.

a) Addition (+):

      Add a number k in each terms of an A.P. , then

a1 + k , a2 + k , a3 + k , … are also in A.P.

b) Subtraction (—) :

      Subtract a number k in each terms of an A.P. , then

a1 – k , a2 – k , a3 – k , … are also in A.P.

c) Multiplication ( x) :

      Multiply a number k in each terms of an A.P. , then

a1 x k , a2 x k , a3 x k , … are also in A.P.

d) Division ( / ):

      Divide a number k in each terms of an A.P. , then

a1 / k , a2 /k , a3/k , … are also in A.P. ( k ≠ 0 ).

2. Construct a series with two A.P’s

If  a1 , a2 , a3 , …… and  b1 , b2 , b3 …. are two A.P’s.

  1. Addition :

      (a1 + b1 ) + (a2 + b2 ) + (a3 + b3 ) + ….. are also in A.P.

b)  Subtraction :

     (a1 – b1 ) + (a2 – b2 ) + (a3 – b3 ) + ….. are also in A.P.

c)   Multiplication :

     (a1 x b1 ) + (a2 x b2 ) + (a3 x b3 ) + ….. are not in A.P.

d) Division :

     (a1 / b1 ) + (a2 / b2 ) + (a3 / b3 ) + ….. are not in A.P.

3 . If  a1 , a2 , a3 , ……  are in A.P.

a )  a1 + an = a2 + an-1 = a3 + an-2 = ……..

b )  ar = 1/2( ar-k + ar+k )  ,    0 ≤ k ≤ n-r

4. Three numbers are taken in A.P .

   a – d , a  , a + d.

similarly , five numbers are taken in A.P .

  a – 2d , a-d , a , a+d , a + 2d . etc .

In general —

(2r + 1) numbers are taken in A.P.

a – rd , a-(r-1)d … , a -d , a , a + d , …, a + (r-1)d , a+rd.

5.  Four  numbers are taken in A.P .

              a – 3d , a – d , a+d , a+3d ;

similarly , six numbers are taken in A.P .

a – 5d , a-3d , a-d , a+d , a+3d , a+5d;

In general —

2r numbers are taken in A.P.

a – (2r-1)d , a-(2r-3)d ,….. , a-3d , a-d , a+d , a , a+3d , …, a +(2r-3)d , a + (2r-1)d .

6. Arithmetic Progression is a linear expression in n.

Ex.  an + b.

7.  Sum of  n terms of any sequence is quadratic expression in n , then sequence in A.P.

Ex .   an2 + bn + c

7. Common difference of an A.P may be zero , positive or negative.