College Algebra
Tutorial 54D: Geometric Sequences and Series
 Learning Objectives
After completing this tutorial, you should be able to:
- Know what a geometric sequence is.
- Find the nth
term of a geometric sequence.
- Write the formula for the nth term of a geometric
sequence.
- Calculate a finite geometric series.
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Introduction
In this tutorial we will mainly be going over geometric
sequences and series. We will be going forwards and
backwards with this. First we will be given the formula for the nth term and we will be
finding specified terms. Then we will turn it around and look at
the terms and find the formula for the nth term. We will finish up
by looking at geometric series. If you need a review on
sequences, feel free to go to Tutorial 54A: Sequences.
If you need a review on sequences, feel free to go to Tutorial 54B: Series. I
think that you are ready to move ahead. |
Tutorial
A geometric sequence is a sequence such that each
successive term is obtained from the previous term by multiplying by a
fixed number called a common ratio.
The sequence 5, 10, 20, 40, 80, .... is an example of a geometric
sequence. The pattern is that we are always multiplying by a
fixed number of 2 to the previous term to get to the next term.
Be careful that you don't think that every sequence that has a pattern
in multiplication is geometric. It is geometric if you are always
multiplying by the SAME number each time.
If you need a review on sequences, feel free to go to Tutorial
54A: Sequences.
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nth
or General Term
of a Geometric Sequence
where  is the first term of the sequence
and r is the common ratio.
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 Example
1: Find the first five terms and the common
ratio of the geometric sequence  .
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Since a geometric sequence is a sequence, you find
the terms exactly the same way that you do a sequence. n is our term number and we
plug the term number into the function to find the value of the term.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
Lets see what we get for
our first five terms:
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What
would be the common ratio for this problem?
If you said 3 you are correct!
Note that you would have to multiply by 3 to each previous term to get
the next term: (15)(3) = 45, (45)(3) = 135, (135)(3) = 405, and
(405)(3) = 1215. It has to be consistent throughout the
sequence.
Also note that the base that is being raised to a power is 3.
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Example 2:
Find the first five terms and the common ratio of the geometric
sequence  .
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Since a geometric sequence is a sequence, you find
the terms exactly the same way that you do a sequence. n is our term number and we
plug the term number into the function to find the value of the term.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
Lets see what we get for
our first five terms:
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What
would be the common ratio for this problem?
If you said -1/2 you are correct!
Note that you would have to multiply by -1/2 each time you go from one
term to the next: (1)(-1/2) = -1/2, (-1/2)(-1/2) = 1/4,
(1/4)(-1/2) = -1/8, and (-1/8)(-1/2)=1/16. It has to be
consistent throughout the sequence.
Also note that the base that is being raised to a power is -1/2.
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Example 3:
Write a formula for the nth
term of the geometric sequence 7, 28, 112, 448, .... Do not use a
recursive formula.
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We will use the nth
term formula for a geometric sequence, to help us with this problem.
Basically we need to find two things: the first term of the
sequence, and the common ratio, r.
What
is , the first term?
If you said 7, give yourself a high five. The first term of this
sequence is 7.
What
is r, the common ratio?
If you said 4, you are right!! Note that you would have to
multiply 4 each time you go from one term to the next: (7)(4) = 28,
(28)(4) = 112, and (112)(4) = 448. It has to be consistent
throughout the sequence.
Putting in 7 for and 4 for r we get:
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Example 4:
Write a formula for the nth
term of the geometric sequence 16, - 4, 1, -1/4, .... Do not use
a recursive formula.
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We will use the nth
term formula for a geometric sequence, to help us with
this problem.
Basically we need to find two things: the first term of the
sequence, and the
common ratio, r.
What
is , the first term?
If you said 16, give yourself a high five. The first term of this
sequence is 16.
What
is r, the common ratio?
If you said -1/4, you are right!! Note that you would have to
multiply -1/4
each time you go from one term to the next: (16)(-1/4) = - 4, (-
4)(-1/4) = 1, and (1)(-1/4) = -1/4. It has to be consistent
throughout the sequence.
Putting in 16 for and -1/4 for r we
get:
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Example 5:
Find the first term of a geometric sequence with a fifth term of 32 and
a common ratio of -2.
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This problem has a little twist to it. Now we are
looking for the first term. We can still use the nth term formula for a
geometric sequence, , to help us with this
problem. We will just be looking for .
Plugging in 32 for nth term, 5 for n, and -2 for r we get:
The first term would have
to be 2.
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Example 6:
Find the common ratio for a geometric sequence with a first term of 3/4
and a third term of 27/16.
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This problem has a little twist to it. Now we are
looking for the common ratio. We can still use the nth term formula for a
geometric sequence, , to help us
with this problem. We will just be looking for r.
Plugging in 3/4 for , 3 for n, and 27/16 for the nth term we get:
The common ratio could be either 3/2 or -3/2.
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The
Sum of the First n Terms of a
Finite Geometric Sequence
is the first term of the
sequence and
r is the common
ratio.
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Example 7:
Find the sum of the finite geometric series 3 - 6 + 12 - 24 + 48 - 96.
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We will use the formula for the sum of the first n terms of geometric
sequence, , to
help us with this problem.
Basically we need to find three things: the first term of the sequence,
the common ratio, and how many terms of the sequence we are adding in
the series.
What
is , the first term?
If you said 3 you are right!
What
is r, the common
ratio?
If you said -2, give yourself a pat on the back. Note that you would
have to multiply -2 each time you go from one term to the next: (3)(-2)
= - 6, (-6)(2) = -12, (-12)(-2) = 24, (24)(-2) = - 48, and (-48)(-4) =
-96. It has to be consistent throughout the sequence.
How
many terms are we summing up?
If you said 6, you are correct.
Putting in 3 for the first
term, -2 for the common ratio, and 6 for n, we get:
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Example 8:
Find the sum of the finite geometric series  .
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We will use the formula for the sum of the first n terms of geometric
sequence, , to
help us with this problem.
Basically we need to find three things: the first term of the sequence,
the common ratio, and how many terms of the sequence we are adding in
the series.
What
is , the first term?
If you said 3 you are right!
Since this summation starts at 0, you need to plug in 0 into the given
formula:
What
is r, the common
ratio?
If you said 1.1, give yourself a pat on the back. Note that 1.1
is the number that is being raised to the exponent. So each time
the number goes up on the exponent, in essence you are multiplying it
by 1.1
How
many terms are we summing up?
If you said 21, you are correct. If you start at 0 and go all the
way to 20, there will be 21 terms.
Putting in 3 for the first
term, 1.1 for the common ratio, and 21 for n, we get:
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The Sum of an Infinite Geometric
Series
If -1 < r < 1 (or  ), then the sum of
the infinite
geometric series
in which is the first term and
r is the common ratio is given by
 .
If  ,
the infinite series does NOT have a sum.
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Example 9:
Find the sum of the infinite series  , if possible.
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We will use the formula for the sum of infinite
geometric sequence, , to help us with this problem.
Basically we need to find two things: the first term of the sequence
and the common ratio.
What
is the first term, ?
If you said 2 you are right!
What
is the common ratio, r?
If you said 1/3, give yourself a pat on the back. Note that you would
have to multiply 1/3 each time you go from one term to the next:
(2)(1/3) = 2/3, (2/3)(1/3) = 2/9, (2/9)(1/3) = 2/27. It has
to be consistent throughout the sequence.
Putting in 2 for the first
term and 1/3 for the common ratio we get:
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Example 10:
Find the sum of the infinite series  , if possible.
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We will use the formula for the sum of infinite
geometric sequence, , to help us with this problem.
Basically we need to find two things: the first term of the sequence
and the common ratio.
What
is the first term, ?
If you said 1.5 you are right!
What
is the common ratio, r?
If you said 2, give yourself a pat on the back. Note that you would
have to multiply 2 each time you go from one term to the next: (1.5)(2)
= 3, (3)(2) = 6, (6)(2) = 12. It has to be consistent
throughout the sequence.
Since the geometric ratio
is 2 and  , there is no sum.
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Example 11:
Find the sum of the infinite series  , if possible.
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We will use the formula for the sum of infinite
geometric sequence, , to help us with this problem.
Basically we need to find two things: the first term of the sequence
and the common ratio.
What
is the first term, ?
If you said -5 you are right!
Since this summation starts at 0, you need to plug in 0 into the given
formula:
What
is the common ratio, r?
If you said -.5, give yourself a pat on the back. Note that -.5 is the
number that is being raised to the exponent. So each time the
number goes up on the exponent, in essence you are multiplying it by
-.5.
Putting in -5 for the
first term and -.5 for the common ratio we get:
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Practice Problems
These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. |
 Practice Problems 1a - 1b: Find the first five terms and
the common ratio of the given geometric sequence.
Practice Problems 2a - 2b: Write a formula for the nth term of the given geometric
sequence. Do not use a recursive formula.
Practice Problems 3a - 3b: Find the sum of the given
finite geometric series.
Practice Problems 4a - 4b: Find the sum of the given
infinite geometric series,
if possible.
Need Extra Help on these Topics?
Last revised on May 17, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.
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