In this explainer, we will learn how to calculate the common ratio, find next terms in a geometric sequence, and check if the sequence increases or decreases.

A sequence is an enumerated collection of numbers (or other objects) that usually follow a pattern. The individual elements in a sequence, for , are called the terms and are labeled by the index , which tells you the position of the given term in the sequence.

There are many real-world applications of geometric sequences in science, business, personal finance, and health. For instance, physicists use geometric sequences to calculate the amount of radioactive material left after any given number of half-lives of the material. During each half-life, the material decays by .

Before we begin with the mathematical description of a geometric sequence, it is useful to consider some simple examples to help us visualize what the structure of a geometric sequence looks like in the real-world. One of the most famous examples of a geometric sequence concerns the invention of chess. According to a legend, Grand Vizier Sissa Ben Dahir invented the game of chess and gifted the Indian King Shaim with a chessboard. As gratitude for the gift, the grand vizier was offered any reward that he requested, as long as it sounded reasonable to the king. The grand vizier made a seemingly humble request with some rice on a chessboard. He requested that one grain of rice be placed on the first square two on the second, four on the fourth, etc, doubling the number of rice grains each time from the previous square until the end of the chessboard (which has 64 squares).

The king, impressed by the modest request, obliged and ordered for the rice to be brought in bags. For the first few squares, everything seemed to be going well, but by the 21st square there were more than a million (1 048 576) grains of rice; the bag was emptied and another had to be brought in, which was immediately emptied by the proceeding square. By the 41st square, there were more than a trillion (1 099 511 627 776) grains of rice and as this progression continued, doubling each time, there were more grains of rice in final squares than exist in the entire world, even without counting all the rice grains on all the previous squares. Just the grains on the final square would exceed the world production of rice for over 1 000 years. The number of rice grains on each square on the chessboard forms a geometric sequence described by where each consecutive term in the sequence is double the previous one; this is known as an increasing geometric sequence.

We can also equivalently depict this as

Now, suppose we want to investigate the behavior of a bouncing tennis ball. If we initially drop the ball from a height of 10 m and measure its position over time, we may notice that the ball loses its original height after each bounce.

The ball loses of its energy every time it rebounds, and the kinetic energy is proportional to the height it was dropped from. This means that the ball loses of its height after every bounce; in other words, each height is of the previous one. This allows us to predict the height of the ball after every bounce. After the first bounce, the height of the ball will be of 10 m:

And after the second bounce, it will be of 8 m:

If we continue this procedure, the height of the ball after each bounce will form a sequence: or equivalently depicted as

This is known as a decreasing geometric sequence, where each term is decreasing and determined by the previous one by multiplying it by a particular number; in this case, the number is 0.8.

Now, let’s define what a geometric sequence is mathematically.

### Definition: Geometric Sequence

A geometric sequence, also known as a geometric progression, is a sequence of nonzero numbers that has a nonzero constant common ratio between any two consecutive terms:

The geometric sequence in general can also be depicted as

For the rice grains on each square of the chessboard, the ratio between any two consecutive terms is , which is the number we multiply each term in the sequence by to get the next term in the sequence.

It is worth noting that geometric sequences do not always increase in value; they can also be decreasing, as with the height of the tennis ball, or alternating in sign. Another example of a decreasing geometric sequence is and an example of an alternating sequence is

The decreasing sequence has a common ratio of and the alternating sequence has a common ratio of . In general, the common ratio controls how the terms in the geometric sequence behave, although we may need to take into account the sign of the starting value, , to know whether it is increasing or decreasing.

- If (equivalent to
or ), the magnitude of the
terms in the geometric sequence, ,
will increase to infinity; this is known as a divergent sequence.
- If and , the terms in the geometric sequence will be increasing and diverging to infinity. For example, with and , the geometric sequence will be:
- If and , the terms in the geometric sequence will be decreasing and diverging to negative infinity. For example, with and , the geometric sequence will be:
- If for any starting value , the terms in the geometric sequence will alternate in sign, but their magnitude will be increasing and diverging to infinity. For example, with and , the geometric sequence will be: or with and , the geometric sequence will be:

- If (equivalent to
), the magnitude of the terms in the geometrics
sequence, , will get closer to
zero; this is known as a convergent sequence.
- If and , the terms in the geometric sequence will be decreasing and converging to zero. For example, with and , the geometric sequence will be:
- If and , the terms in the geometric sequence will be increasing and converging to zero. For example, with and , the geometric sequence will be:
- If for any starting value , the terms in the geometric sequence will be alternating in sign but their magnitude will be decreasing and converging to zero. For example, with and , the geometric sequence will be: or with and , the geometric sequence will be:

In order to calculate the common ratio of a given geometric sequence, we can divide any term from the sequence by the term that immediately precedes it. For example, we could divide the third term by the second term or the second term by the first term in the sequence; either way should yield the same number for a geometric sequence.

In the first example, we will determine the common ratio for a given geometric sequence describing the number of bacteria in a laboratory experiment.

### Example 1: Finding the Common Ratio of a Geometric Sequence

The table shows the number of bacteria in a laboratory experiment across four consecutive days. The number of bacteria can be described by a geometric sequence. Find the common ratio of this sequence.

Day | 1st | 2nd | 3rd | 4th |
---|---|---|---|---|

Number of Bacteria | 643 | 2 572 | 10 288 | 41 152 |

### Answer

In this example, we want to determine the common ratio of a sequence describing the number of bacteria in a laboratory experiment.

Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. As we are told that we are looking at a geometric sequence, we do not need to check that the ratio is consistent between every two consecutive terms. Instead, we can choose any two consecutive terms. The common ratio can be found from the 2nd and 1st term as

We could have also found this from the ratio of the 3rd and 2nd terms or the 4th and 3rd terms to yield the same result. We also note that since and , the geometric sequence is increasing and divergent.

Thus, the common ratio of the geometric sequence is 4.

In the last example, we determine the common ratio from the 2nd and 1st terms, but we could have also used the 3rd and 2nd or 4th and 3rd terms to yield the same result: or

We could also be asked to find successive terms in a geometric sequence. This involves, first, finding the common ratio and then using it to calculate additional terms by multiplying the preceding terms by .

Now, let’s consider an example where we find the next term of a given decreasing geometric sequence.

### Example 2: Finding the Next Term of a Given Geometric Sequence

Find the next term of the geometric sequence .

### Answer

In this example, we want to determine the next term of a given geometric sequence.

Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain

We note that, since and , the geometric sequence is increasing and converging to zero.

Finally, we can determine the next term in the sequence, , by multiplying the term before it by to obtain

In the last example, we saw that we can determine the next term of a given sequence by first finding the common ratio and multiplying the last term by it. If we repeat this procedure, we can enumerate as many terms in the sequence as required.

In the next example, we will determine the next four terms of a given increasing geometric sequence.

### Example 3: Finding the Missing Terms of a Given Geometric Sequence

Find the next four terms in the geometric sequence .

### Answer

In this example, we want to determine the next four terms in a given geometric sequence.

Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. The first step is to establish the common ratio of the geometric sequence, which we can find from the ratio of any two consecutive terms. We can use the ratio of the 2nd and 1st terms in the sequence to obtain

We note that since and , the geometric sequence is increasing and divergent. The next four terms, , can be found by repeatedly multiplying the previous terms by this common ratio:

Thus, the next four terms in the geometric sequence are

As we may note from the definition and previous examples, the recursive formula for the geometric sequence can be written as

In some cases, we may be given a geometric sequence as a relation in this form, which we can use to determine the common ratio.

In the next example, we will identify the common ratio of a given geometric sequence defined in terms of a recursive formula.

### Example 4: Finding the Common Ratio of a Geometric Sequence given Its General Term

Find the common ratio of the geometric sequence that satisfies the relation , where .

### Answer

In this example, we want to determine the common ratio of a geometric sequence defined by a particular recursive relation.

Recall that a sequence is geometric if there is a common ratio, , between any two consecutive terms:

We can also rearrange this recursive formula for the geometric sequence as

We note that this is different from the form given in the question where the ratio is multiplying . Now, if we rewrite the given relation in this form, we obtain

Thus, the common ratio is

We note that, since , this geometric sequence is converging to zero; whether it increases or decreases depends on the starting value .

If we are also given the starting value, , with the recursive formula, we can enumerate the first few terms in the sequence; this is the same thing as previous examples where we multiply the common ratio with a term in the sequence to get the next term.

Now, let’s consider an example where we have to find the first five terms of a given geometric sequence and a starting value.

### Example 5: Finding the Terms of a Sequence given Its General Term and the Value of the First Term

Find the first five terms of the sequence , given , , and .

### Answer

In this example, we want to find the terms of a geometric sequence from a given relation and the first term.

We note that, since and from the given recursive formula as , the geometric sequence is increasing and converging to zero.

We begin by finding the 2nd term by substituting into the relation:

Then, we repeat this procedure to find the other three terms with :

Thus, the first five terms of the sequence are

It is also possible to do things in the opposite way around to the last example. That is, supposing we have the first few terms of a sequence, we can find its common ratio and first term. In fact, if we denote the first term as for simplicity, the general form of a geometric sequence is

The second term of the geometric sequence is calculated by multiplying the first term, , by to obtain , then the third term is the second term multiplied by , to give , and so on.

In other words, every term is multiplied by the same number, , to produce the next term. In the next example, we will see an application of this idea.

### Example 6: Identifying the First Term and Common Ratio of a Convergent Geometric Sequence

A geometric sequence is a list of terms that can be written in the form where is the first term and is the common ratio (the number you multiply one term by to get the next term in the sequence, ).

Identify and in the following sequence: .

### Answer

In this example, we want to identify the first term, , and the common ratio, , of a given geometric sequence.

The first term by inspection is . The second term, from the given form, is . Substituting the value of into the second expression and solving for gives us the common ratio, which we can also determine from any two consecutive terms:

We note that, since and , the geometric sequence is decreasing and converging to zero.

Thus, we have

Much like how we can use the properties of a geometric sequence to determine the ratio or specific terms, we can verify whether a sequence is geometric or not by checking the properties it has. In the following example, we will see an application of this.

### Example 7: Identifying Geometric Sequences

Which of the following is not a geometric sequence?

### Answer

In this example, we will need to check each option to see whether it correctly follows the definition of a geometric sequence.

Recall that a geometric sequence is defined as one that has a common ratio, , between successive terms. That is,

At first, it may seem daunting trying to identify which sequences (some of which appear to be quite complicated) obey this principle. However, there is a straightforward procedure we can follow in each case to verify the sequences; namely, we can divide the second term by the first term to obtain the ratio, and then we can multiply the second term by this ratio to see whether we get the third term or not. If we do obtain it, then the ratio is a common ratio and the sequence is geometric.

Thus, let us apply this procedure. For option A, is and is , so the ratio is

Next, we multiply the second term by this ratio:

Since is indeed , this confirms that the ratio is consistent between the first three terms. Additionally, we can check and confirm that it is equal to . Thus, option A is a geometric sequence.

For option B, the ratio between and is

Then, we multiply by the ratio to get which is . Furthermore, (i.e., ), which shows that option B is also geometric.

For option C, the ratio between the first two terms is where we have used the property to simplify the expression. However, if we multiply by , we get

This is not equal to ; rather, it is actually . Thus, the sequence cannot be geometric, since the ratio between terms is not constant.

For completeness, let us consider option D. We can calculate the ratio between the first two terms to be

Calculating this ratio multiplied by , we get

This is equal to . We can also verify that . Thus, this sequence is geometric.

In conclusion, option C is the only nongeometric sequence.

We can also determine the values of unknown parameters appearing in the terms of a geometric sequence by using the properties of geometric sequences, in particular the common ratio being constant for any two consecutive terms.

Now, let’s consider an example where we are given three terms in a geometric sequence given in terms of an unknown parameter, which we will determine from the properties of geometric sequences.

### Example 8: Using the Properties of Geometric Sequences to Find an Unknown Term

Find the value of given the geometric sequence .

### Answer

In this example, we will use the properties of a geometric sequence in order to determine an unknown parameter appearing in the terms of the sequence.

Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. As we are told we have a geometric sequence, we know that the ratio between any two consecutive terms is a constant number . From the 2nd and 1st terms, we have while from the 3rd and 2nd terms, we have

These ratios will be the same for a geometric sequence and we will obtain the quadratic equation

Thus, the solution is or . We note that these solutions give rise to different types of geometric sequences. For , the common ratio is and, since and , we have a decreasing and divergent geometric sequence. For , the common ratio is and, since and , this is an increasing and convergent geometric sequence.

In the previous example, we saw that an unknown parameter appearing in the terms of a geometric sequence had two solutions, each with a different type of geometric sequence, either increasing or decreasing.

In the final example, we will determine the value of two unknown parameters appearing in two of the four terms in a given geometric sequence using its properties.

### Example 9: Using the Properties of Geometric Sequences to Find the Values of Unknown Terms

Find and given the geometric sequence .

### Answer

In this example, we will use the properties of a geometric sequence in order to determine the unknown parameters and appearing in the terms of the sequence.

Recall that a sequence is geometric if there is a common ratio between any two consecutive terms. As we are told we have a geometric sequence, we know that the ratio between any two consecutive terms is a constant number . From the 4th and 3rd terms, we have while, from the 3rd and 2nd terms, we have and from the 2nd and 1st terms we have

These ratios will be the same for a geometric sequence and we will obtain the quadratic equations

Upon rearranging, these equations become

Substituting the second equation into the third yields the quadratic equation

Thus, we have and , but we can ignore the first solution since it does not satisfy the first equation . Using the first equation, the value of can be found as .

Therefore, the values of and that appear in the geometric sequence are

We note that, for this solution, the common ratio is . And since and , we have an increasing and divergent geometric sequence.

Let’s finish by summarizing the key points we have learned in this explainer.

### Key Points

- A geometric sequence is a sequence of nonzero numbers defined by a constant nonzero common ratio between any two consecutive terms in the sequence A given geometric sequence may be defined in terms of a set of numbers , a recursive formula, or an explicit formula.
- In general, the common ratio controls how the terms in
the geometric sequence behave, although we may need to take into account the
sign on the starting value, , to know whether it is
increasing or decreasing.
- If (equivalent to
or
), the magnitude of the terms in the geometric
sequence, , will increase to
infinity; this is known as a divergent sequence.
- If and , the terms in the geometric sequence will be increasing and diverging to infinity.
- If and , the terms in the geometric sequence will be decreasing and diverging to negative infinity.
- If for any starting value , the terms in the geometric sequence will alternate in sign but their magnitude will be increasing and diverging to infinity.

- If (equivalent to
), the magnitude of the terms in the
geometrics sequence, , will get
closer to zero; this is known as a
convergent sequence.
- If and , the terms in the geometric sequence will be decreasing and converging to zero.
- If and , the terms in the geometric sequence will be increasing and converging to zero.
- If for any starting value , the terms in the geometric sequence will be alternating in sign but their magnitude will be decreasing and converging to zero.

- If (equivalent to
or
), the magnitude of the terms in the geometric
sequence, , will increase to
infinity; this is known as a divergent sequence.
- A recursive relation, which follows from the definition of the common ratio, along with a given starting value, , allows us to enumerate the terms in a geometric sequence: This means that each term in a geometric sequence is found by multiplying the previous term by the common ratio.
- If we denote the starting value as for simplicity, the general form of a geometric sequence is
- Using the properties of geometric sequences, we can also determine the values of unknown parameters appearing in some of the terms of a given geometric sequence.