# nth Term of an AP

#### description

• general term of the AP

#### notes

nth term of an AP is also known as last term or general term.
Let a be the first term of an AP and d be the common difference then a standard form of an AP will be a, a+d, a+2d, a+3d,............. and so on upto nth term.
In an AP terms can also be written as a_1, a_2, a_3, a_4,........, a_n
a_1= a+ (1-1)d
a_2= a+d= a+ (2-1)d
a_3= a+2d=  a+ (3-1)d
a_4= a+3d= a+ (4-1)d
And so on...
So if we generalize this pattern we get
a_n= a+ (n-1)d
Example- In an AP 2, 7, 12, ........... Then find a20
a= 2, n=20, d= a_2-a_1= 7-2= 5
a_20= 2+ (20-1)5
= 2+ 19(5)
= 2+ 95
a_20= 97
Now, what will do if we are asked find nth term form the end? So, we will use a different approch.
Let a be the first term of an AP and d be the common difference then a standard form of an AP will be a, a+d, a+2d, a+3d, ......, l  where l is a last term.
If the last term is l then the term before l will be l-d, and if the second last term is l-d then l-d-d  i.e l-2d will be the term before it and so on.
First term from the end= l= l- (1-1)d
Second term from the end= l-d= l- (2-1)d
Third term form the end= l-2d= l- (3-1)d
And so on...
So if we generalize this pattern we get
nth term from the end= l- (n-1)d
Example- 4, 9, 14, ....., 254 Find the 10th term from the end
a= 4, n=10, d= a_2-a_1= 9-4= 5, l=254
nth term from the end= l- (n-1)d
10th term from the end= 254- (10-1)5
= 254- (9)5
= 254-45
10th term from the end= 209

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