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.

ALGEBRA
examples of geometric sequences

Relevant for

Exploring examples with answers of geometric sequences.

See examples

ALGEBRA
examples of geometric sequences

Relevant for

Exploring examples with answers of geometric sequences.

See examples

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:

geometric sequences example

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

formula of geometric sequences

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

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, ?.

We have the variable x and we have constant terms. Therefore, we combine with terms with variables and combine the constant terms:

2x+4+3x-5

=(2x+3x)+(4-5)

=5x-1

EXAMPLE 2

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

We have a parenthesis so we start by using the distributive property to distribute the 2 and remove the parentheses:

3x+2(3x-2)+10

=3x+6x-4+10

Now we combine like terms. We combine the variables and the constant terms:

=(3x+6x)+(-4+10)

=9x+6

EXAMPLE 3

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

In this case, we have the variable x with a power of 1 and with a power of 2, so we combine like terms separately for the power of 1 and for the power of 2. We also combine the constant terms separately:

4x+2{{x}^2}+5-3x+4{{x}^2}+4

=(2{{x}^2}+4{{x}^2})+(4x-3x)+(5+4)

=4{{x}^2}+x+9

EXAMPLE 4

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

We start by applying the distributive property to remove the parentheses:

3x(2x+5)+10x-6+5{{x}^2}

=6{{x}^2}+15x+10x-6+5{{x}^2}

Now combine like terms. We combine terms with the variable x with different powers separately:

=(6{{x}^2}+5{{x}^2})+(15x+10x)-6

=11{{x}^2}+25x-6

EXAMPLE 5

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

We start by removing the parentheses using the distributive property:

4x(y+3x)-3xy+5{{x}^2}+10-6x

=4xy+12{{x}^2}-3xy+5{{x}^2}+10-6x

In this case, we have terms with both the x variable and the y variable. We have to combine terms that have the same variables raised to the same powers:

=(4xy-3xy)+(12{{x}^2}+5{{x}^2})+10-6x

=xy+17{{x}^2}-6x+10

EXAMPLE 6

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

We start by removing both parentheses using the distributive property:

5(2{{x}^2y+4x)+3x-2x(4xy-5)+12

=10{{x}^2}y+20x+3x-8{{x}^2}y-10x+12

We have to combine the terms that have the same variables raised to the same power:

=(10{{x}^2}y-8{{x}^2}y)+(20x+3x-10x)+12

=2{{x}^2}y+13x+12

EXAMPLE 7

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

We start by removing the parentheses using the distributive property:

-2x(3{{x}^2}+2x-1)+4x+2{{x}^2}+6-4{{x}^3}-10x+5

=-6{{x}^3}-4{{x}^2}+2x+4x+2{{x}^2}+6-4{{x}^3}-10x+5

Here we have several terms with the variable x with different powers. We have to make sure to combine only the terms that have the same power:

=(-6{{x}^3}-4{{x}^3})+(-4{{x}^2} +2{{x}^2})+(2x+4x-10x)+(5+6)

=-10{{x}^3}-2{{x}^2}-4x+11


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.

Find the next term in the sequence: 3, 9, 27, 81, ?.

Choose an answer






What is the next term in the sequence? 240, 120, 60, 30, ?.

Choose an answer






Find the term 8 in the sequence: 6, 12, 24, 48, …

Choose an answer






Find term 7 in the sequence: 540, 220, 110, 55, …

Choose an answer






Find term 9 in the sequence: 2, 6, 18, 54…

Choose an answer







See also

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