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nth Term of an AP

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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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