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Finding the n th Term of an Arithmetic Sequence

Given an arithmetic sequence with the first term a 1 and the common difference d , the n th (or general) term is given by a n = a 1 + ( n 1 ) d .

Example 1:

Find the 27 th term of the arithmetic sequence  5 , 8 , 11 , 54 , ... .

a 1 = 5 , d = 8 5 = 3

So,

a 27 = 5 + ( 27 1 ) ( 3 ) = 83

Example 2:

Find the 40 th term for the arithmetic sequence in which
a 8 = 60 and a 12 = 48 .

Substitute 60 for a 8 and 48 for a 12 in the formula
a n = a 1 + ( n 1 ) d  to obtain a system of linear equations in terms of a 1 and d .

a 8 = a 1 + ( 8 1 ) d 60 = a 1 + 7 d a 12 = a 1 + ( 12 1 ) d 48 = a 1 + 11 d

Subtract the second equation from the first equation and solve for d .

12 = 4 d 3 = d

Then 60 = a 1 + 7 ( 3 ) .  Solve for a .
60 = a 1 21 81 = a 1

Now use the formula to find a 40 .

a 40 = 81 + 39 ( 3 ) = 81 117 = 36 .   

See also: sigma notation of a series and n th term of a geometric sequence