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Algebra II: nth term of an A.P. from the end
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nth term of an A.P. from the end 

Let there be an A.P. with first term a and common difference d. If there are m terms in the A.P., then
nth term from the end  = (m - n + l)th term the beginning
 = am-n+l
 = a + (m - n + 1 - 1)d
 = a + (m - n)d

Also, if l is the last term of the A.P., then nth term from the end is the nth term of an A.P. whose first term is l and common difference is -d.
Thus,
            nth term from the end  = Last term + (n -1) (- d)
 = l - (n - 1)d

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Example: Which term of the A.P. 2, 7, 12, 17......... will be 60 more than its 55th term?
Solution:
Given A.P. = 2, 7, 12, 17....
So
First term (a) = 2 and common difference (d) = 5.
Let the nth term of the A.P. be 60 more than its 55th term i.e.
an = 60 + a55
a + (n-1)d = 60 + (a + 54d)
2 + (n-1)5 = 60 + (2 + 54 x 5)
5n - 3 = 332
5n = 335
n = 67
Hence, 67th term of the given A.P. is 60 more than its 55th term.
 
   
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