Arithmetic Progression(AP)

        Basic Of Arithmetic Progression(AP)

First let's  remind Some basic Definitions:

Arithmetic Progression: It is the sequence of numbers such that the difference between the consecutive terms is constant.

Examples: 

1,2,3,4,5,6,7,8.  here 2-1=3-2=4-3......same every where so in AP.

2,4,6,8,10,12,14. here 4-2=6-4=8-6......also same every where  so in AP.

etc.

 

Term: Each number in Series is called Term. 

example: here 1,2,3,4,5,6,7,8 are Terms in AP

Common Difference: The term which is Constant in Series called Common Difference.

example:

1,2,3,4,5,6,7,8.  here 2-1=3-2=4-3......difference same every where so in AP. (Common Difference=> 2-1=1.)
her in all case  difference is same and equal to 1. so Common Difference(cd)=1.

2,4,6,8,10,12,14. here 4-2=6-4=8-6......difference also same every where  so in AP.
her in all case  difference is same and equal to 2. si Common Difference(cd)=2.
etc.

 Note: Common Difference(cd) may be Positive, Negative or Zero. 

   General Form of Arithmetic Progression          

  Let's say a  is First Term d  is Common Difference then AP(Arithmetic Progression) can be defined as...

a, a+d, a+2d, a+3d, a+4d,..........

Finite Arithmetic Progression(AP): In AP if the number of terms are finite then called Finite Arithmetic Progression(AP).

Examples:

1,2,3,4,5,6,7,8........20.  here mentioned that you have to go up to 20 only, so finite Number.

2,4,6,8,10. here already mentioned finitely.

 Term (General Term): Sometimes it is also called Last Term, since we can find Last Term by using General Term Formula(). 

 =a+(n-1)d

here,

	=	the nᵗʰ term in the sequence

a = First term of an AP,

n = No of terms in an AP,

d = Common Difference,

Let's take an Example: 2,4,6,8,10,12,....20.

a= 2,  

n=10,

d=4-2=6-4=8-6.....=2,

=ℓ=20.

We can find anything term by using the formula, if Series is in AP.

 =a+(n-1)d

Infinite Arithmetic Progression(AP): In AP if the number of terms are Infinite then called Infinite Arithmetic Progression(AP).

Examples:

1,2,3,4,5,6,7,8.........  here not mentioned that you have to go up to what place, so Infinite Number.

2,4,6,8,10........ here also not mentioned last term.

Last Term in AP:  As the name suggest Last Term, means the Last number in AP is called Last Term.

Example: 

2,4,6,8,10,12,14.................30.  Here Last Term is 30.

This is generally denoted by  ℓ (

एल).

here  ℓ=30.

How to Find Summation  Of  AP(s) Series?

Let's say a  First Term d  Common Difference and  is nᵗʰ term   then AP(Arithmetic Progression) can be defined as...

a, a+d, a+2d, a+3d, a+4d,...........

S= sum of AP.

S= n/2[2a + (n – 1)d]

or 

S=n/2[a + l]

l = Last term of an AP.

Some important formulas for adding Specific Numbers

Sum of 'n' term in AP:                                        n/2[2a + (n – 1)d]

Sum of 'n' natural Numbers:                             n(n+1)/2

Sum of square of 'n' Natural numbers:           [n(n+1)(2n+1)]/6

Sum of Cube of 'n' Natural numbers:            [n(n+1)/2]2

Thanks!

Happy Learning :)