ARITHMETIC PROGRESSION

ARITHMETIC PROGRESSION (AP)

Arithmetic progression in which the difference between the consecutive term is constant.

In other words the next term is calculated by adding a fixed number in the previous term. This fixed term is called common difference.

For example:

3,6,9,12.....is an arithmetic progression because the common difference is same i.e. ⟦ 6-3=9-6=12-9=3⟧ 

General representation of AP:

a,a+d,a+2d,a+3d.........

Here a is called the first term and d is the common difference.

nth term= a+(n-1)d

Arithmetic mean= sum of terms of AP/No of terms of AP

Sum of AP (S)=n/2⟦2a+(n-1)⟧d

If the a is first term and T is the last term then

 Sum of AP (S)=n/2⟦a+T⟧

Example:

1. Find the nth term of 1,3,5,7.....
    nth term=a+(n-1)d
                  =1+(n-1)2

                   =1+2n-2
     nth term=2n-1

 2. Find the 10th term of 2,4,6,8,10...
     for this sequence a=2
       and common difference=2
      nth term=a+(n-1)d
      10th term=2+(10-1)2
                      =2+9*2
       10th term =20

3.Find the no of term in the 8,12,16..........72
  for this sequence a=8
    and common difference=4
  No. of term=(l-a)/d+1
                     =(72-8)/4+1
                      =64/4+1
                       =16+1
                       =17
No of term=17

4. Find 3+7+11+15.....+20 terms
  For this a=3
and common difference=4
Sum= n/2⟦2a+(n-1)⟧d
        =20/2[2*3+(20-1)]4
         =10[6+19]4
          =40*25
    Sum   =1000