What if you had a series of numbers like 1, 3, 9, 27 ... in which each number was found by multiplying the previous number by a fixed amount? How could you find the fifth and sixth numbers in this series? After completing this Concept, you'll be able to solve problems like this one that involve geometric sequences.
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For more practice finding the terms in geometric sequences, try the browser game at http://www.iesmath.com/math/java/misc/sum/sum.html.
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CK12 Foundation: 0810S Lesson Geometric Sequences
Guidance
Consider the following question:
Which would you prefer, being given one million dollars, or one penny the first day, double that penny the next day, and then double the previous day's pennies and so on for a month?
At first glance it’s easy to say "Give me the million!" But why don’t we do a few calculations to see how the other choice stacks up?
You start with a penny the first day and keep doubling each day. Doubling means that we keep multiplying by 2 each day for one month (30 days).
On the first day, you get 1 penny, or
On the second day, you get 2 pennies, or
On the third day, you get 4 pennies, or
On the fourth day, you get 8 pennies, or
So on the thirtieth day, you get
This problem is an example of a geometric sequence. In this section, we’ll find out what a geometric sequence is and how to solve problems involving geometric sequences.
Identify a Geometric Sequence
A geometric sequence is a sequence of numbers in which each number in the sequence is found by multiplying the previous number by a fixed amount called the common ratio. In other words, the ratio between any term and the term before it is always the same. In the previous example the common ratio was 2, as the number of pennies doubled each day.
The common ratio,
Here are some examples of geometric sequences and their common ratios.
If we know the common ratio
Example A
Fill in the missing terms in each geometric sequence.
a) 1, ___, 25, 125, ___
b) 20, ___, 5, ___, 1.25
Solution
a) First we can find the common ratio by dividing 125 by 25 to obtain
To find the first missing term, we multiply 1 by the common ratio:
To find the second missing term, we multiply 125 by the common ratio:
Sequence (a) becomes: 1, 5, 25, 125, 625,...
b) We need to find the common ratio first, but how do we do that when we have no terms next to each other that we can divide?
Well, we know that to get from 20 to 5 in the sequence we must multiply 20 by the common ratio twice: once to get to the second term in the sequence, and again to get to five. So we can say
Dividing both sides by 20, we get
To get the first missing term, we multiply 20 by
To get the second missing term, we multiply 5 by
Sequence (b) becomes: 20, 10, 5, 2.5, 1.25,...
You can see that if we keep multiplying by the common ratio, we can find any term in the sequence that we want—the tenth term, the fiftieth term, the thousandth term.... However, it would be awfully tedious to keep multiplying over and over again in order to find a term that is a long way from the start. What could we do instead of just multiplying repeatedly?
Let’s look at a geometric sequence that starts with the number 7 and has common ratio of 2.
The nth term is
The formula for finding a specific term in a geometric sequence is:
(
Example B
For each of these geometric sequences, find the eighth term in the sequence.
a) 1, 2, 4,...
b) 16, 8, 4, 2, 1,...
Solution
a) First we need to find the common ratio:
The eighth term is given by the formula
In other words, to get the eighth term we start with the first term, which is 1, and then multiply by 2 seven times.
b) The common ratio is
The eighth term in the sequence is
Let’s take another look at the terms in that second sequence. Notice that they alternate positive, negative, positive, negative all the way down the list. When you see this pattern, you know the common ratio is negative; multiplying by a negative number each time means that the sign of each term is opposite the sign of the previous term.
Solve RealWorld Problems Involving Geometric Sequences
Let’s solve two application problems involving geometric sequences.
Example C
A courtier presented the Indian king with a beautiful, handmade chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, etc. The king readily agreed and asked for the rice to be brought. (From Meadows et al. 1972, via Porritt 2005) How many grains of rice does the king have to put on the last square?
Solution
A chessboard is an
The courtier asked for one grain of rice on the first square, 2 grains of rice on the second square, 4 grains of rice on the third square and so on. We can write this as a geometric sequence:
1, 2, 4,...
The numbers double each time, so the common ratio is
The problem asks how many grains of rice the king needs to put on the last square, so we need to find the
The problem we just solved has real applications in business and technology. In technology strategy, the Second Half of the Chessboard is a phrase, coined by a man named Ray Kurzweil, in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.
The total number of grains of rice on the first half of the chessboard is
The total number of grains of rice on the second half of the chessboard is
Watch this video for help with the Examples above.
CK12 Foundation: Geometric Sequences
Vocabulary
 A geometric sequence is a sequence of numbers in which each number in the sequence is found by multiplying the previous number by a fixed amount called the common ratio. In other words, the ratio between any term and the term before it is always the same. In the previous example the common ratio was 2, as the number of pennies doubled each day.
 The common ratio,
r , in any geometric sequence can be found by dividing any term by the preceding term.
Guided Practice
A superball has a 75% rebound ratio—that is, when it bounces repeatedly, each bounce is 75% as high as the previous bounce. When you drop it from a height of 20 feet:
a) how high does the ball bounce after it strikes the ground for the third time?
b) how high does the ball bounce after it strikes the ground for the seventeenth time?
Solution
We can write a geometric sequence that gives the height of each bounce with the common ratio of
a) The ball starts at a height of 20 feet; after the first bounce it reaches a height of
After the second bounce it reaches a height of
After the third bounce it reaches a height of
b) Notice that the height after the first bounce corresponds to the second term in the sequence, the height after the second bounce corresponds to the third term in the sequence and so on.
This means that the height after the seventeenth bounce corresponds to the
Here is a graph that represents this information. (The heights at points other than the top of each bounce are just approximations.)
Explore More
Determine the first five terms of each geometric sequence.

a1=2,r=3 
a1=90,r=−13 
a1=6,r=−2 
a1=1,r=5 
a1=5,r=5 
a1=25,r=5  What do you notice about the last three sequences?
Find the missing terms in each geometric sequence.
 3, __ , 48, 192, __
 81, __ , __ , __ , 1

94 ,−− ,−− ,23 ,−−  2, __ , __ , 54, 162
Find the indicated term of each geometric sequence.

a1=4,r=2 ; finda6 
a1=−7,r=−34 ; finda4 
a1=−10,r=−3 ; finda10  In a geometric sequence,
a3=28 anda5=112; findr anda1 .  In a geometric sequence,
a2=28 anda5=112; findr anda1 .  As you can see from the previous two questions, the same terms can show up in sequences with different ratios.
 Write a geometric sequence that has 1 and 9 as two of the terms (not necessarily the first two).
 Write a different geometric sequence that also has 1 and 9 as two of the terms.
 Write a geometric sequence that has 6 and 24 as two of the terms.
 Write a different geometric sequence that also has 6 and 24 as two of the terms.
 What is the common ratio of the sequence whose first three terms are 2, 6, 18?
 What is the common ratio of the sequence whose first three terms are 18, 6, 2?
 What is the relationship between those ratios?
 Anne goes bungee jumping off a bridge above water. On the initial jump, the bungee cord stretches by 120 feet. On the next bounce the stretch is 60% of the original jump and each additional bounce the rope stretches by 60% of the previous stretch.
 What will the rope stretch be on the third bounce?
 What will be the rope stretch be on the
12th bounce?