What is the common ratio of a geometric sequence whose 3rd term is − 9 and whose 8th term is 2187?

Given a geometric sequence with the first term a 1 and the common ratio r , the n th (or general) term is given by
a n = a 1 ⋅ r n − 1 .

Example 1:

Find the 6 th term in the geometric sequence 3 , 12 , 48 , ... .

a 1 = 3 ,     r = 12 3 = 4 a 6 = 3 ⋅ 4 6 − 1 = 3 ⋅ 4 5 = 3072

Example 2:

Find the 7 th term for the geometric sequence in which a 2 = 24 and a 5 = 3 .

Substitute 24 for a 2 and 3 for a 5 in the formula

a n = a 1 ⋅ r n − 1 .

a 2 = a 1 ⋅ r 2 − 1 → 24 = a 1 r a 5 = a 1 ⋅ r 5 − 1 →         3 = a 1 r 4

Solve the firstequation for a 1 : a 1 = 24 r

Substitute this expression for a 1 in the second equation and solve for r .

3 = 24 r ⋅ r 4 3 = 24 r 3 1 8 = r 3   so   r = 1 2

Substitute for r in the first equation and solve for a 1 .

24 = a 1 ( 1 2 ) 48 = a 1

Now use the formula to find a 7 .

a 7 = 48 ( 1 2 ) 7 − 1 = 48 ⋅ 1 64 = 3 4

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