ARITHMETIC PROGRESSION : How to find Nth term of the series ?

UTPAL SHARMA

Arithmetic Progression 

1.) Definitions used in Arithmetic Progression (A.P.)


Arithmetic Progression ( A.P.):- 
A series of numbers in which common difference (the difference between two successive terms) are same.

Common Difference ( d ) :- 
Difference between two successive terms of an A.P.
OR
A number which we add in preceding term & get succeeding term.

Number of terms (n):- 
It is the number of terms in any A.P.  or position of any term.

Nth term (an) or (l) :- 
It is a term of an A.P. for n position. Where n may be 1,2,3,4....etc. It may be the last term in some A.P.

Last term (l) or (an) :- 
As per the name, it is the last term of an A.P.

General Form of A.P.:-
a , a+d , a+2d, a+3d......a+(n-1)d.

Where a = first term & d = common difference

All the above terms discussing in the below picture.

In this image we explained details of general form of Arithmetic Progression.


2.) Basics of Arithmetic Progression (A.P.)

1.) Series of A.P. may be negative, positive or both.

2.) Common difference (d) may be negative, positive or zero in A.P.

3.) When we add the common difference in any term of A.P. then we get its next term.

4.) When we subtract the common difference in any term of A.P. then we get its previous term.

5.) To find the value of d, subtract preceding term from succeeding term.

6.) To find the unknown terms in A.P. we have a formula - an = a + (n-1)d

3.) Example of Arithmetic Progression(A.P.)

You may compare the below picture from the above general form for better understanding.
In image we are discussing an example of Arithmetic Progression.


4.) How to find an unknown term in Arithmetic Progression(A.P.)?

We understand it with the help of an example. For this we have a formula, to find an unknown term or Nth term in an A.P.
an = a + (n-1)d

Find the twelfth term of an A.P. If the first term is 2 & the common difference is 2.

Given-
a=2, d=2, n=12 (Because we are solving for twelfth term so n=12) & a12 =?

Now put all the values in formula…
  •  an = a + (n-1)d
  • a12 = 2 + (12-1)2
  • a12 = 2 + (11)2
  •  a12 = 2 + 22
  •  a12 = 24
So our twelfth term in given A.P. will be a12 = 24.

In Arithmetic Progression we can find the position of any term. 

Means, the value of n & also we may find the value of common difference d for the given term.

Let's discuss-

5.) How to find the position of the known term in Arithmetic Progression(A.P.)?

Find the value of n for the term 30 in the given A.P.-
2,4,6,8,.........

  •  Given- an = 30 , a = 2 , d = 2 (4 - 2) , n=?
  •  an = a + (n-1)d
  •  30 = 2 + (n-1)2
  •   30 - 2 = (n-1)2  
  •  28 = (n-1)2
  •   28 ÷ 2 = (n-1)
  •   14 = n-1
  •   14 +1 = n
  •   15 = n
Above solution shows number 30 is placed on fifteenth position so n = 15 for a15 = 30.

According to this, we may solve for common difference d.
 
So we learned how we may find any unknown term in A.P. & its position.

We may find the sum of the first nth term of an A.P. & other.
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