Name: _____________________
Math 12R
Date: __________
Geometric Sequence / Series
Aim: What are Geometric Sequences and Series?
Geometric Sequence
a1
10
•
•
a2
2
a3
0.4
a4
0.08
a5
0.016
The sequence given in the table above is an example of an geometric sequence. Notice
that each successive term is not found by adding a constant.
The ratio of successive terms in a geometric sequence is a constant called the common
ratio, r.
A geometric sequence is a sequence in which each term (except the first term) is the product of
the preceding term and the common ratio, r.
The terms of the sequence can be represented as follows, where a1 is nonzero and r is not
equal to 1 or 0.
a1, a1 r, a1 r2, a1 r3,...
The common ratio of a geometric sequence can be found by dividing any term by the
preceding term. Then multiply the last term by the common ratio to find the next term in
the sequence.
Example: Find the next three terms in the geometric sequence 27, 135, 675, …
The nth Term of an Geometric Sequence:
an = a1 · r (n – 1)
a1 = 1st term
r = common ratio
n = number of terms
Example:
Find an approximation for the 23rd term in the geometric sequence 256, -179.2, 125. 44, …
Example:
Find the second and third terms of a geometric sequence whose first term is 48 and fourth
term is -750.
Geometric Series
A geometric series is the indicated sum of the terms of a geometric sequence.
Examples:
Geometric Sequence
1, 4, 16, 64, 256
2,1,
1 1 1
, ,
2 4 8
a1, a2, a3, a4, …,an
Geometric Series
1 + 4 + 16 + 64 + 256
2 1
1
2
1
4
1
8
a1 + a2 + a3 + a4 + …+ an
The Symbol Sn is used to represent the sum of the first n terms of a series. Rather than adding
all the terms from a1 to an, we have a formula for Sn for an geometric series. It is:
Sum of an Geometric Series:
Sn
a1 a1r n
1 r
Sn = Sum of the first n terms
a1 = 1st term
r = common ratio
Example: Find the sum of the first 8 terms of the series: 3
6 + 12+…
Practice
Determine the common ratio and find the next three terms in each sequence.
(1.) 36, -12, 4, …
(2.) 2, 3, …
(3.) 7, 3.5,…
(4.) The first term of a geometric sequence is -3, and the common ratio is
terms.
2
. Find the next four
3
(5.) Find the sum of the first ten terms of the geometric series 16 – 48 + 144 – 432 + …
(6.) The first term of a geometric sequence is
1
2
and r = . Find the 9th term of the sequence.
2
3
(7.) If r = 2 and a5 = 24, find the first term of the geometric sequence.
(8.) Form a sequence which has three geometric means between 2 and
(9.) Find the ninth term of the geometric sequence
1
.
8
2, 2, 2 2, ...
(10.) Form a sequence which has two geometric means between -2 and 54.
Name: _____________________
Math 12R
Date: __________
Geometric Sequence / Series - HW
In problems 1-3, determine the common ratio, and then find the next two terms of each
geometric sequence.
1.) 10, 2, 0.4, …
2.) 8, -20, 50, …
3.) t8, t5, t2, …
4.) Find the 5th term of a sequence whose first term is 8 and common ratio is 1.5
5.) Find the sixth term of the sequence
1 3 9
,
, ,...
2 8 32
6.) If r = 4 and a6 = 192, what is the first term of this sequence?
7.) What is the sum of the first six terms of the series 65 + 13 + 2.6 + … ?