The nth term of a geometric sequence can be found by multiplying the previous term by a number called the common ratio. Then, we can form a formula to find any term of a geometric sequence using the value of an initial term, the common ratio, and the position of the required term.

Here, we will look at the formula that we can use to find the nth term of a geometric sequence. Then, we will apply this formula to solve some examples.

## Steps to find the nth term of a geometric sequence

Consider the sequence of numbers 2, 6, 18, 54, … Each term in this sequence can be obtained from the previous term by multiplying it by 3. This is an example of a geometric sequence.

Geometric sequences are sequences of numbers in which each term can be obtained by multiplying the previous term by a number called the **common ratio**.

Generally, the terms of a geometric sequence can be obtained using the following formula:

$$a_{n}=ar^{n-1}$$

where,

- $latex a$ is the first term of the sequence.
- $latex r$ is the common ratio.
- $latex n $ is the position of the term.

Therefore, we can find the nth term of a geometric sequence with the following steps.

#### 1. Identify the value of the first term.

#### 2. Find the value of the common ratio.

The common ratio is found by dividing any term by its previous term.

#### 3. Apply the formula for the nth term.

Use the values of the first term, the common ratio, and the position of the term in the formula $latex a_{n}=ar^{n-1}$.

## Solved examples of the nth term of a geometric sequence

**EXAMPLE **1

**EXAMPLE**

If a geometric sequence has a first term equal to 2 and the common ratio is 3, find the value of the 5th term.

##### Solution

We know the values of the first term and the common ratio directly. Then, we start by writing all the known values:

- $latex a=2$
- $latex r=3$
- $latex n=5$

Now, we can apply the formula for the nth term of a geometric sequence:

$latex a_{n}=ar^{n-1}$

$latex a_{5}=(2)(3)^{5-1}$

$latex a_{5}=(2)(3)^4$

$latex a_{5}=(2)(81)$

$latex a_{5}=162$

**EXAMPLE **2

**EXAMPLE**What is the value of the 9th term of a geometric sequence in which its first term is 5 and the common ratio is 2?

##### Solution

Similar to the previous example, we already know the values of the first term and the common ratio directly. Then, we have:

- $latex a=5$
- $latex r=2$
- $latex n=9$

Using these values in the formula for the nth term of a geometric sequence, we have:

$latex a_{n}=ar^{n-1}$

$latex a_{9}=(5)(2)^{9-1}$

$latex a_{9}=(5)(2)^8$

$latex a_{9}=(5)(256)$

$latex a_{9}=1280$

**EXAMPLE **3

**EXAMPLE**Find the 7th term of a geometric sequence that begins with the terms 4, 8, 16, and 32.

##### Solution

In this case, we don’t know the value of the common ratio directly, but we do know the values of the first four terms.

Therefore, we can find the common ratio by dividing one of the terms by its previous term. For example, $latex \frac{8}{4}=2$. Then, we have:

- $latex a=4$
- $latex r=2$
- $latex n=7$

Using the formula for the nth term, we have:

$latex a_{n}=ar^{n-1}$

$latex a_{7}=(4)(2)^{7-1}$

$latex a_{7}=(4)(2)^6$

$latex a_{7}=(4)(64)$

$latex a_{7}=256$

**EXAMPLE **4

**EXAMPLE**What is the 7th term of a geometric sequence that begins with terms 1, -3, and 9?

##### Solution

Similar to the previous example, we can find the common ratio by dividing any term by its previous term. For example, $latex \frac{9}{-3}=-3$. Then, we have:

- $latex a=1$
- $latex r=-3$
- $latex n=7$

In this case, we have a negative common ratio, but we simply apply the formula for the nth term:

$latex a_{n}=ar^{n-1}$

$latex a_{7}=(1)(-3)^{7-1}$

$latex a_{7}=(1)(-3)^6$

$latex a_{7}=(1)(729)$

$latex a_{7}=729$

**EXAMPLE **5

**EXAMPLE**Find the 8th term of a geometric sequence that starts with the terms 81, -54, and 36.

##### Solution

The common ratio of this sequence is $latex \frac{36}{-54}=-\frac{2}{3}$. Therefore, we have:

- $latex a=81$
- $latex r=-\frac{2}{3}$
- $latex n=8$

Using these values in the formula for the nth term, we have:

$latex a_{n}=ar^{n-1}$

$$a_{8}=(81)\left(-\frac{2}{3}\right)^{8-1}$$

$$a_{8}=(81)\left(-\frac{2}{3}\right)^7$$

$$a_{8}=(81)\left(-\frac{128}{2187}\right)$$

$$a_{8}=-\frac{10368}{2187}$$

$$a_{8}=-\frac{128}{27}$$

$$a_{8}=-4~\frac{20}{27}$$

**EXAMPLE **6

**EXAMPLE**A geometric sequence has the first three terms $latex 1,~-\frac{1}{3}, ~\frac{1}{9}$. Find the 6th term.

##### Solution

The common ratio of the given sequence is $latex \frac{1}{9} \div -\frac{1}{3}=-\frac{1}{3}$. Thus, we have:

- $latex a=1$
- $latex r=-\frac{1}{3}$
- $latex n=6$

Using the formula for the nth term with these values, we have:

$latex a_{n}=ar^{n-1}$

$$a_{6}=(1)\left(-\frac{1}{3}\right)^{6-1}$$

$$a_{6}=(1)\left(-\frac{1}{3}\right)^5$$

$$a_{6}=(1)\left(-\frac{1}{243}\right)$$

$$a_{6}=-\frac{1}{243}$$

You can explore more examples on this topic in our article: nth Term of an Geometric Sequence – Examples and Practice.

## nth Term of geometric sequences – Practice problems

#### Find the 100th term of a geometric sequence that begins with the terms 7, -7, 7, …

Write the answer in the input box.

## See also

Interested in learning more about geometric sequences? You can take a look at these pages:

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