Geometric sequences have the main characteristic of having a common ratio, which is multiplied by the last term to find the next term. Any term in a geometric sequence can be found using a formula.

Here, we will look at a summary of geometric sequences and we will explore its formula. In addition, we will see several examples with answers and exercises to solve to practice these concepts.

## Summary of geometric sequences

Geometric sequences are sequences in which the next number in the sequence is found by multiplying the previous term by a number called the **common ratio**. The common ratio is denoted by the letter *r*.

Depending on the common ratio, the geometric sequence can be increasing or decreasing. If the common ratio is greater than 1, the sequence is increasing and if the common ratio is between 0 and 1, the sequence is decreasing:

We can find any number in the geometric sequence using the geometric sequence formula:

We can find the common ratio by dividing any term by the previous term:

$latex r=\frac{a_{n}}{a_{n-1}}$

## Geometric sequences – Examples with answers

The following examples of geometric sequences have their respective solution. The solutions show the process to follow step by step to find the correct answer. It is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

Find the next term in the geometric sequence: 4, 8, 16, 32, **?**.

##### Solution

First, we have to find the common ratio of the geometric progression. To do this, we divide a term by the previous term:

- $latex \frac{32}{16}=2$
- $latex \frac{16}{8}=2$
- $latex \frac{8}{4}=2$

Therefore, the common ratio is 2. To find the next term, we multiply the last term by the common ratio: $latex 32\times 2=64$.

**EXAMPLE 2**

What is the next term in the geometric sequence? 3, 15, 75, 375, **?**.

##### Solution

We start by finding the common ratio for the geometric progression. Then, we divide each term by its previous term:

- $latex \frac{375}{75}=5$
- $latex \frac{75}{15}=5$
- $latex \frac{15}{3}=5$

We see that the common ratio is 5. We find the next term by multiplying the last term by the common ratio : $latex 375 \times 5=1875$.

**EXAMPLE 3**

Determine the next term in the geometric sequence: 48, 24, 12, 6, **?**.

##### Solution

Again, we start by finding the common ratio in the progression:

- $latex \frac{6}{12}=0.5$
- $latex \frac{12}{24}=0.5$
- $latex \frac{24}{48}=0.5$

In this case, we see that the common ratio is between 0 and 1, so the progression is slowing down. The next term in the geometric progression is $latex 6\times 0.5=3$.

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**EXAMPLE 4**

Find the term 12 in the geometric sequence: 5, 15, 45, 135, …

##### Solution

In this case, we have to use the formula of geometric progressions $latex a_{n}=a_{1}({{r}^{n-1}})$. Therefore, we have to identify the first term, the common reason and the position of the term:

- First term: $latex a_{1}=5$
- Common ratio: $latex r=3$
- Position of term: $latex n=12$

Now, we substitute this data into the formula:

$latex a_{n}=a_{1}({{r}^{n-1}})$

$latex a_{12}=5({{3}^{12-1}})$

$latex a_{12}=5({{3}^{11}})$

$latex a_{12}=5(177147)$

$latex a_{12}=885 735$

We see that we have a very large number. Geometric progressions tend to grow rapidly depending on the common proportion.

**EXAMPLE 5**

Find the term 8 in the geometric sequence 8, 32, 128, 512, …

##### Solution

Again, we start by identifying the first term, the common ratio, and the position of the term to be used with the formula:

- First term: $latex a_{1}=8$
- Common ratio: $latex r=4$
- Position of term: $latex n=8$

Now, we use the formula with these values:

$latex a_{n}=a_{1}({{r}^{n-1}})$

$latex a_{8}=8({{4}^{8-1}})$

$latex a_{8}=8({{4}^{7}})$

$latex a_{8}=8(16384)$

$latex a_{8}=131072$

**EXAMPLE 6**

Find the term 10 in the geometric sequence: 168, 84, 42, 21, …

##### Solution

In this case, we have a decreasing geometric progression, so we expect the common ratio to be between 0 and 1:

- First term: $latex a_{1}=168$
- Common ratio: $latex r=0.5$
- Position of term: $latex n=10$

We use the formula to find the term 10:

$latex a_{n}=a_{1}({{r}^{n-1}})$

$latex a_{10}=168({{0.5}^{10-1}})$

$latex a_{10}=168({{0.5}^{9}})$

$latex a_{10}=168(0.001953)$

$latex a_{10}=0.328$

**EXAMPLE 7**

Find the term 7 in the geometric sequence: 540, 180, 60, 20, …

##### Solution

Similar to the previous example, here we have a decreasing geometric progression, so the common ratio must be between 0 and 1:

- First term: $latex a_{1}=540$
- Common ratio: $latex r=\frac{1}{3}$
- Possiion of term: $latex n=7$

We use these values to substitute in the formula:

$latex a_{n}=a_{1}({{r}^{n-1}})$

$latex a_{7}=540({{\left( \frac{1}{3}\right)}^{7-1}})$

$latex a_{7}=540({{\left( \frac{1}{3}\right)}^{6}})$

$latex a_{7}=540(0.0013717)$

$latex a_{7}=0.7407$

## Geometric sequences – Practice problems

Practice and test your knowledge of geometric sequences with the following problems. Select one of the options and check it to see if you got the correct answer. You can guide yourself with the solved examples above if you have problems.

## See also

Interested in learning more about sequences? Take a look at these pages:

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