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Geometric Sequences
Another simple way of generating a sequence is to start with a number “a” and repeatedly
multiply it by a fixed nonzero constant “r”. This type of sequence is called a geometric
sequence.
Definition: A geometric sequence is a sequence of the form
234
aa, r, ar, ar, ar,...
The number a is the first term, and r is the common ratio of the sequence. The
nth term of a geometric sequence is given by
n−1
aa= r.
n
The number r is called the common ratio because any two consecutive terms of the sequence
differ by a multiple of r, and it is found by dividing any term a after the first by the preceding
n+1
term a . That is
n
a
r = n+1 .
a
n
Is the Sequence Geometric?
Example 1: Determine whether the sequence is geometric. If it is geometric, find the common
ratio.
(a) 2,8,32,128,...
(b) 1, 2, 3, 5, 8, ...
Solution (a): In order for a sequence to be geometric, the ratio of any term to the
one that precedes it should be the same for all terms. If they are all
the same, then r, the common difference, is that value.
Step 1: First, calculate the ratios between each term and the one that precedes it.
8 = 4
2
32 = 4
8
128 = 4
32
By: Crystal Hull
Example 1 (Continued):
Step 2: Now, compare the ratios. Since the ratio between each term and the one
that precedes it is 4 for all the terms, the sequence is geometric, and the
common ratio r=4.
Solution (b):
Step 1: Calculate the ratios between each term and the one that precedes it.
2 =1
1
33
=
22
55
=
33
88
=
55
Step 2: Compare the ratios. Since they are not all the same, the sequence is not
geometric.
Similar to an arithmetic sequence, a geometric sequence is determined completely by the first
term a, and the common ratio r. Thus, if we know the first two terms of a geometric sequence,
then we can find the equation for the nth term.
Finding the Terms of a Geometric Sequence:
th
Example 2: Find the nth term, the fifth term, and the 100 term, of the geometric sequence
determined by 1
ar==6, .
3
Solution: To find a specific term of a geometric sequence, we use the formula
for finding the nth term.
Step 1: The nth term of a geometric sequence is given by
n−1
aa= r
n
So, to find the nth term, substitute the given values 1 into the formula.
ar6,
= = 3
1 n−1
a =6⎛⎞
n ⎜⎟
3
⎝⎠
By: Crystal Hull
Example 2 (Continued):
Step 2: Now, to find the fifth term, substitute n = 5 into the equation for the nth
term.
1 51−
a =6⎛⎞
5 ⎜⎟
3
⎝⎠
1
=6⎛⎞
⎜⎟
4
3
⎝⎠
= 6
81
= 2
27
Step 3: Finally, find the 100th term in the same way as the fifth term.
1 100−1
a =6⎛⎞
5 ⎜⎟
3
⎝⎠
1
=6⎛⎞
⎜⎟
99
3
⎝⎠
= 23⋅
99
3
= 2
98
3
Example 3: Find the common ratio, the fifth term and the nth term of the geometric sequence.
(a) −−1, 9, 81, 729,...
23
1 tt t
(b) 2, 6, 18, 54,...
Solution (a): In order to find the nth term, we will first have to determine what a
and r are. We will then use the formula for finding the nth term of
a geometric sequence.
By: Crystal Hull
Example 3 (Continued):
Step 1: First, determine what a and r are. The number a is always the first term
of the sequence, so
a = −1.
The ratio between any term and the one that precedes it should be the same
because the sequence is geometric, so we can choose any pair to find the
r. If we choose the first two terms
common ratio
r = 9
−
1
=−
9.
Step 2: Since we are given the fourth term, we can multiply it by the common
ratio r=−9 to get the fifth term.
= ⋅
aar
54
=729 −9
()
=−6561
Step 3: Now, to find the nth term, substitute ar= −=1, −9 into the formula for
the nth term of a geometric sequence.
n−1
aa= r
n
=− −n−1
19
()()
−
=−− n1
9
()
By: Crystal Hull
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