Finding the n th Term of a Geometric Sequence

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 n1 .

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 61 =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 n1 .

a 2 = a 1 r 21 24= a 1 r a 5 = a 1 r 51 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 ) 71 =48 1 64 = 3 4

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