A **geometric sequence** is a type of numeric sequence that increases or decreases by a constant multiplication or division.

A geometric sequence is also sometimes referred to as a *geometric progression*.

Each number in a sequence is referred to as a **term**.

Geometric sequences can help you calculate many things in real life, such as:

Finances – compound interest.

Population growth.

Decay.

1, 3, 9, 27, 81, ... is a geometric sequence: every number in the sequence is obtained by multiplying the previous number by 3.

In this case, 3 is the so-called* common ratio* of the geometric sequence: let's find out more about it.

## Geometric sequence common ratio

If you are given a term of a geometric sequence you can find the following term by multiplying the initial term by a constant, known as the **common ratio**. This procedure is known as the **term to term rule**.

The common ratio is often denoted as *r*.

Some examples of geometric sequences include:

- 3, 6, 12, 24, 48... This sequence has a common ratio of 2 since each term is obtained by multiplying the previous one by 2.
- 5, 20, 80, 320, 1280... This sequence has a common ratio of 4 since each term is obtained by multiplying the previous one by 4.
- 32, 16, 8, 4, 2, 1, 0.5... This sequence has a common ratio of 0.5 since each term is obtained by multiplying the previous one by 0.5.

### How to find the common ratio in a geometric sequence

When you are given a geometric sequence, you may not be given the common ratio. It can be helpful to be able to figure this out in case you need to find the next terms of the geometric sequence. **To find the common ratio you shall divide one term by the term before it.**

Find the common ratio for the geometric sequence 6, 18, 54, 162, 486...

**Solution:**

To find the common ratio of this geometric sequence, take the second term and divide it by the first term of the sequence. To check your result, divide the third term by the second; the fourth by the third; and so on.

$18\xf76=3$

$54\xf718=3$

$162\xf754=3$

$486\xf7162=3$

Therefore the common ratio for this geometric sequence is 3.

## nth term of geometric sequence

It is possible to use the** term to term rule** to find the nth terms of a geometric sequence. To do this multiply or divide the term you have by the common ratio to find the next term of the sequence.

Find the next three terms of the geometric sequence 8, 40, 200, 1000...

**Solution:**

First, you need to identify the common ratio:

$40\xf78=5$

To make sure that the common ratio is 5, check the following terms:

$200\xf740=5$

$1000\xf7200=5$

Now you know that the common ratio is 5, you can use that to find the next terms of the sequence. Just multiply the last term by the common ratio and repeat that to find the next three terms:

$1000\times 5=5000$

$5000\times 5=25000$

$25000\times 5=125000$

Therefore, the next three terms of the sequence are 5000, 25000, 125000

Since you are multiplying the terms, the terms will rapidly increase or decrease.

Find the first five terms of the geometric sequence where the first term is 13 and the common ratio is 2.

**Solution:**

To find each term you can start by multiplying the first term by the term to term rule:

$13\times 2=26$

Now you can continue to multiply the term to term rule by the previous term:

$26\times 2=52$

$52\times 2=104$

$104\times 2=208$

Therefore the first five terms of the sequence are 13, 26, 52, 104, and 208.

Find the first three terms of the geometric sequence where 1000 is the first term and the common ratio is $\frac{1}{4}$.

**Solution:**

To do this you need to multiply each term by $\frac{1}{4}$ to find the next:

$1000\times \frac{1}{4}=250$

$250\times \frac{1}{4}=62.5$

Therefore the first three terms are; 1000, 250, 62.5

## Difference between arithmetic and geometric sequence

The difference between an arithmetic sequence and a geometric sequence is the way in which the terms go from one to another. In an **arithmetic** **sequence** the terms increase or decrease by a constant addition or subtraction. In a **geometric** **sequence** the terms increase or decrease by a constant multiplication or division.

## Geometric sequence examples with solutions

Identify the common ratio in the following geometric sequence: 11, 33, 99, 297...

**Solution:**

To find the common ratio divide the second term of the sequence by the first term of the sequence and so on:

$33\xf711=3$

$99\xf733=3$

$297\xf799=3$

Therefore, the term to term rule of this sequence is 3.

Find the next 3 terms of the geometric sequence 9, 18, 36, 72, 144…

**Solution:**

First, identify the common ratio:

$18\xf79=2$

$36\xf718=2$

$72\xf736=2$

$144\xf772=2$

Now find the next terms by multiplying the common ratio by the previous term;

$144\times 2=288$

$288\times 2=576$

$576\times 2=1152$

Therefore, the next three terms of the sequence are; 288, 576, and 1152.

Find the first five terms of the geometric sequence where 5 is the first term and the common ratio is 4.

**Solution:**

To do this you need to multiply each term by 4 to find the next:

$5\times 4=20$

$20\times 4=80$

$80\times 4=320$

$320\times 4=1280$

Therefore the first five terms of the sequence are; 5, 20, 80, 320, and 1280.

Identify the term to term rule in the following geometric sequence; 100, 80, 64, 51.2

**Solution:**

To find the common ratio divide one term by the previous term in the sequence on so on:

$80\xf7100=0.8or\frac{4}{5}$

$64\xf780=0.8or\frac{4}{5}$

$51.2\xf764=0.8or\frac{4}{5}$

Therefore, the term to term rule of this sequence is 0.8 or $\frac{4}{5}$.

## Geometric Sequences - Key takeaways

- A geometric sequence is a numerical sequence that increases or decreases by a constant multiplication.
- The constant ratio between each term in the sequence is called the
**common ratio.** - The common ratio can be used to generate terms of the sequence.

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##### Frequently Asked Questions about Geometric Sequence

What is the geometric sequence?

A geometric sequence is a type of linear sequence that increases or decreases by a constant multiplication or division.

How do you find arithmetic and geometric sequences?

To find the difference between an arithmetic and geometric sequence you must find out how the sequences are increasing or decreasing;

- If the sequence is increasing or decreasing by a constant addition or subtraction, it is an arithmetic sequence.
- If the sequence is increasing or decreasing by a constant multiplication or division it is a geometric sequence.

What is a finite geometric sequence?

A finite geometric sequence is a geometric sequence that has an end.

What is sum to infinity of a geometric sequence?

The sum to infinity of a geometric sequence is when all the terms in the sequence are added together.

How to find nth term of geometric sequence?

In order to find the nth term of a geometric sequence you can multiply or divide the last term with the common ratio in order to find the next term.

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