Geometric Sequences

Geometric Sequences

A geometric sequence is a sequence in which each successive term is formed by multiplying the same number. In a geometric sequence, the ratio between consecutive terms is constant. This constant is called the common ratio and is denoted with the letter, r.

Example 1 - Which of the following sequences are geometric? For those that are, state the common ratio.

Sequence	Ratio Between Successive Terms	Geometric or Not?	i	Sequence	Ratio Between Successive Terms	Geometric or Not?
-6, 12, 24, -48, ...	12/(-6) = -2 24/12 = 2 -48/24 = -2	This sequence is not geometric.	i		-8/(-32) = -2/(-8) = -2/(-)= (-2)(-4) = 8	This sequence is not geometric.
0.2, 0.4, 0.6, 0.8,...	0.4/0.2 = 2 0.6/0.4 = 1.5 0.8/0.6 = 1.3...	This sequence is not geometric	i			This sequence is geometric. r = 2
	-1/6 = - ()/(-1) = - (-)/()= -(6)= -	This sequence is geometric. r = -	i			This sequence is geometric. r =

Formula for the General Term of an Geometric Sequence

                                  where = the nth term
                                                             a = first term of the sequence
                                                              r = common ratio
                                                             n = number of terms or term number

Example 2 For each sequence, find the indicated term.
a)	the 25th term of 2, -4, 8, ...	b)	the 12th term of
	Solution The sequence is geometric with a first term of 2 and a common ratio of -4/2 = -2. Thus, a = 2 and r = -2.		Solution The sequence is geometric with a first term of 6 and a common ratio of . Thus, a = 6 and r = .
Example 3 For each geometric sequence, write a formula for the nth term and then use it to find the 15th term.
a)	1, -5, 25, ...	b)
	Solution The sequence is geometric with a first term of 1 and a common ratio of -5/1 = -5. Thus, a = 1 and r = -5.		Solution The sequence is geometric with a first term of 4 and a common ratio of . Thus, a = - and r = -.
Example 4 Identify how many terms are in the given geometric sequence.
a)	-3, -12, -48, ..., -12,288	b)	48, -12, 3, ...,
	Solution The sequence is geometric with a first term of -2 and a common ratio of -12/-3 = 4. Thus, a = -3 and r = 4.		Solution The sequence is geometric with a first term of 48 and a common ratio of -12/48 = -. Thus, a = 48 and r = -.