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GCSE Maths Algebra Sequences

Geometric Sequences

Geometric Sequences

Here we will learn what a geometric sequence is, how to continue a geometric sequence, how to find missing terms in a geometric sequence, and how to generate a geometric sequence.

At the end, you’ll find geometric sequence worksheets based on Edexcel, AQA, and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a geometric sequence?

A geometric sequence (geometric progression) is an ordered set of numbers that progresses by multiplying or dividing each term by a common ratio.

If we multiply or divide by the same number each time to make the sequence, it is a geometric sequence.

The common ratio is the same for two consecutive terms. Here are a few examples,

First TermTerm-to-Term RuleFirst 5 Terms
3 Multiply by 3 3, 9, 27, 81, 243, ...
5 Multiply by 2 5, 10, 20, 40, 80, ...
-12 Divide by 2 -12, -6, -3, -1.5, -0.75, ...
0.8 Multiply by 5 0.8, 4, 20, 100, 500, ...
\frac{1}{2} Divide by 4 \frac{1}{2}, \; \frac{1}{8}, \; \frac{1}{32}, \; \frac{1}{128}, \; \frac{1}{512} \ldots

What are geometric sequences?

What are geometric sequences?

Geometric sequence formula

The geometric sequence formula is,

geometric sequence formula

Where, 

\pmb{ a_{n} } is the n^{th} term (general term),

\pmb{ a_{1} } is the first term,

\pmb{ n } is the term position,

and \pmb{ r } is the common ratio.

We get the geometric sequence formula by looking at the following example,

Geometric Sequences formula image 1

We can see the common ratio (r) is 2 , so r = 2 .

a_{1} is the first term which is 5 ,

a_{2} is the second term which is 10 ,

and a_{3} is the third term which is 20 etc.

However we can write this using the common difference of 2 ,

Geometric Sequences formula image 2

To find out more about the different types of sequences, and how to answer sequence related questions you may find it helpful to look at the introduction to sequences lesson or one of the others in this section.

How to continue a geometric sequence

To continue a geometric sequence, you need to calculate the common ratio. This is the factor that is used to multiply one term to get the next term. To calculate the common ratio and continue a geometric sequence you need to:

  1. Take two consecutive terms from the sequence.
  2. Divide the second term by the first term to find the common ratio r.
  3. Multiply the last term in the sequence by the common ratio to find the next term. Repeat for each new term.

Explain how to continue a geometric sequence

Explain how to continue a geometric sequence

Geometric sequences worksheet

Geometric sequences worksheet

Geometric sequences worksheet

Get your free geometric sequences worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Geometric sequences worksheet

Geometric sequences worksheet

Geometric sequences worksheet

Get your free geometric sequences worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Continuing a geometric sequence examples

Example 1: continuing a geometric sequence

Calculate the next three terms for the geometric progression 1, 2, 4, 8, 16,

  1. Take two consecutive terms from the sequence.

Here we will take the numbers 4 and 8 .

2 Divide the second term by the first term to find the value of the common ratio, r .

\begin{aligned} r&=8 \div 4 \\\\ r&=2 \end{aligned}

3 Multiply the last term in the sequence by the common ratio to find the next term. Repeat for each new term.

\begin{aligned} 16 \times 2&=32 \\\\ 32 \times 2&=64 \\\\ 64 \times 2&=128 \end{aligned}

The next three terms in the sequence are 32, 64, and 128 .

Example 2: continuing a geometric sequence with negative numbers

Calculate the next three terms for the sequence -2, -10, -50, -250, -1250, …

Here we will take the numbers -10 and -50 .

r=-50 \div -10

r=5

-1250 \times 5=-6,250

-6250 \times 5=-31,250

-31250 \times 5=-156,250

The next three terms are -6,250, -31,250, and -156,250.

Example 3: continuing a geometric sequence with decimals

Calculate the next three terms for the sequence 100, 10, 1, 0.1, 0.01,

Here we will take the numbers 0.1 and 0.01 .

\begin{aligned} r&=0.01 \div 0.1 \\\\ r&=0.1 \end{aligned}

\begin{aligned} 0.01 \times 0.1&=0.001\\\\ 0.001 \times 0.1&=0.0001\\\\ 0.0001 \times 0.1&=0.00001\\\\ \end{aligned}


The next three terms in the sequence are 0.001, 0.0001, and 0.00001.

Example 4: continuing a geometric sequence involving fractions

Calculate the next three terms for the sequence

10, \quad 5, \quad 2\frac{1}{2}, \quad 1 \frac{1}{4}, \quad \frac{5}{8}, ...

Here we will take the numbers 5 and 2\frac{1}{2} .

\begin{aligned} r&=2 \frac{1}{2} \div 5\\\\ r&= \frac {1}{2} \end{aligned}

\begin{aligned} \frac{5}{8} \times \frac{1}{2}&=\frac{5}{16} \\\\ \frac{5}{16} \times \frac{1}{2}&=\frac{5}{32} \\\\ \frac{5}{32} \times \frac{1}{2}&=\frac{5}{64} \end{aligned}


The next three terms are


\frac{5}{16}, \frac{5}{32}, and \frac{5}{64} .

 Geometric sequence practice questions – continue the sequence

1. Write the next three terms of the sequence 0.5, 5, 50, 500, …

1000, 5000, 10000
GCSE Quiz False

950, 1400, 1850
GCSE Quiz False

5000, 50000, 500000
GCSE Quiz True

550, 555, 560
GCSE Quiz False

Choose two consecutive terms. For example, 5 and 50 .

 

Common ratio,

 

50\div 5=10

 

500 \times 10=5000, 5000 \times 10=50000, 50000 \times 10=500000

2. Write the next three terms of the sequence 0.04, 0.2, 1, 5, 25, …

50, 100, 200
GCSE Quiz False

125, 625, 3125
GCSE Quiz True

50, 1000, 20000
GCSE Quiz False

12.5, 6.25, 3.125
GCSE Quiz False

Choose two consecutive terms. For example, 5 and 25 .

 

Common ratio,

 

25\div 5=5

 

25 \times 5=125, 125 \times 5=625, 625 \times 5=3125

3. Calculate the next 3 terms of the sequence -1, -3, -9, -27, -81, …

243, 729, 2187
GCSE Quiz False

-243, -729, -2187
GCSE Quiz True

243, -729, 2187
GCSE Quiz False

-27, -9, -3
GCSE Quiz False

Choose two consecutive terms. For example, -27 and -9 .

 

Common ratio,

 

-27\div -9=3

 

-81 \times 35=-243, -243 \times 3=-729, -729 \times 3=-2187

4. By finding the common ratio, state the next 3 terms of the sequence 640, 160, 40, 10, 2.5 .

0.625, 0.15625, 0.0390625
GCSE Quiz True

1.25, 0.625, 0.3125
GCSE Quiz False

10, 40, 160
GCSE Quiz False

0.25, 0.025, 0.0025
GCSE Quiz False

Choose two consecutive terms. For example, 40 and 10 .

 

Common ratio,

 

10\div 40=\frac{1}{4} \text{ or } 0.25

 

2.5 \times 0.25=0.625, 0.625 \times 0.25=0.15625, 0.15625 \times 0.25=0.0390625

5. Work out the common ratio and therefore the next three terms in the sequence 36, 12, 4, \frac{4}{3}, \frac{4}{9}, …

\frac{12}{9}, \frac{36}{9}, \frac{108}{9}
GCSE Quiz False

\frac{12}{27}, \frac{36}{81}, \frac{108}{243}
GCSE Quiz False

\frac{4}{12}, \frac{4}{15}, \frac{4}{18}
GCSE Quiz False

\frac{4}{27}, \frac{4}{81}, \frac{4}{243}
GCSE Quiz True

Choose two consecutive terms. For example, 12 and 4 .

 

Common ratio,

 

4\div 12=\frac{1}{3}

 

\frac{4}{9} \times \frac{1}{3}=\frac{4}{27}, \frac{4}{27} \times \frac{1}{3}=\frac{4}{81}, \frac{4}{81} \times \frac{1}{3}=\frac{4}{243}

6. Find the common ratio and hence calculate the next three terms of the sequence 1, -1, 1, -1, 1, …

1, 1, 1
GCSE Quiz False

-1, -1, -1
GCSE Quiz False

1, -1, 1
GCSE Quiz False

-1, 1, -1
GCSE Quiz True

Choose two consecutive terms. For example, -1 and 1 .

 

Common ratio,

 

-1\div 1=-1

 

1 \times -1=-1, -1 \times -1=1, 1 \times -1=-1

How to find missing numbers in a geometric sequence

The common ratio can be used to find missing numbers in a geometric sequence. To find missing numbers in a geometric sequence you need to:

  1. Calculate the common ratio between two consecutive terms.
  2. Multiply the term before any missing value by the common ratio.
  3. Divide the term after any missing value by the common ratio.

Repeat Steps 2 and 3 until all missing values are calculated. You may only need to use Step 2 or 3 depending on what terms you have been given.

Explain how to find missing numbers in a geometric sequence

Explain how to find missing numbers in a geometric sequence

Finding missing numbers in a geometric sequence examples

Example 5: find the missing numbers in the geometric sequence

Fill in the missing terms in the sequence 7, 14, …, …,112 .

\begin{aligned} r&=14 \div 7 \\\\ r&=2 \end{aligned}

14 \times 2=28

112 \div 2 = 56


The missing terms are 28 and 56 .


Note: Here, you could repeat Step 2 by using 28 \times 2 = 56.

Example 6: find the missing numbers in a geometric sequence including decimals

Find the missing values in the sequence 0.4, …, ..., 137.2, 960.4.

\begin{aligned} r&=960.4 \div 137.2 \\\\ r&=7 \end{aligned}

0.4 \times 7 = 2.8

137.2 \div 7 = 19.6


The missing terms are 2.8 and 19.6 .

Example 7: find the missing numbers in a geometric sequence when there are multiple consecutive terms missing

Find the missing values in the sequence, -4, ..., …, -108,...

First, we need to find the factor between the two terms, -108 \div -4 = 27 .

To get from -108 to -4 , we jump 3 terms.


This means that -4 has been multiplied by the common ratio three times or -4 \times r \times r \times r = -4r^3 .


\begin{aligned} r^{3}&=27\\\\ r&=3 \end{aligned}

\begin{aligned} -4 \times 3&=-12 \\\\ -12 \times 3&=-36 \\\\ -36 \times 3&=-108 \\\\ -108 \times 3&=-324 \end{aligned}

Note: Term -108 is already given.


The missing terms are -12, -36, and -324.

We don’t need to complete this step.

Example 8: find the missing numbers in a geometric sequence including mixed numbers

Find the missing values in the sequence

…, 4 \frac{1}{2}, \quad 13 \frac{1}{2},\quad ... , \quad...

\begin{aligned} r&=13 \frac{1}{2} \div 4 \frac{1}{2}\\\\ r &=\frac{27}{2} \div \frac{9}{2}\\\\ r &=\frac{27}{2} \times \frac{2}{9}\\\\ r &=3 \end{aligned}

13 \frac{1}{2} \times 3=40 \frac{1}{2}


Repeat this step to find the next term.


40 \frac{1}{2} \times 3=121 \frac{1}{2}

4 \frac{1}{2} \div 3=1 \frac{1}{2}


The missing terms in the sequence are


1 \frac{1}{2}, 40 \frac{1}{2}, and 121 \frac{1}{2} .

 Geometric sequence practice questions – find missing numbers

1. Find the missing numbers in the geometric sequence 4, 2, …, 0.5, …

1, 0.2
GCSE Quiz False

1, 0.25
GCSE Quiz True

1, 0.1
GCSE Quiz False

1, 0
GCSE Quiz False

Choose two consecutive terms. For example, 4 and 2 .

 

Common ratio,

 

2\div 4=0.5

 

2 \times 0.5=1, 0.5 \times 0.5=0.25

2. Find the missing numbers in the sequence -7, -35, …, …, -4375

-175, -875
GCSE Quiz True

175, 875
GCSE Quiz False

-4375, -546875
GCSE Quiz False

1481.7, 2928.3
GCSE Quiz False

Choose two consecutive terms. For example, -7 and -35 .

 

Common ratio,

 

-35\div -7=5

 

-35 \times 5=-175, -175 \times 5=-875

3. Find the missing terms in the sequence 0.6, …, …, 0.075, 0.0375

0.2, 0.067
GCSE Quiz False

1.2, 2.4
GCSE Quiz False

0.12, 0.24
GCSE Quiz False

0.3, 0.15
GCSE Quiz True

Choose two consecutive terms. For example, 0.075 and 0.0375 .

 

Common ratio,

 

0.0375\div 0.075=0.5

 

-35 \times 5=-175, -175 \times 5=-875

4. Calculate the missing terms in the arithmetic sequence 2 \frac{1}{5}, \frac{11}{20}, \frac{11}{80}, \ldots, \ldots

\frac{11}{320}, \frac{11}{1280}
GCSE Quiz True

\frac{44}{80}, \frac{176}{80}
GCSE Quiz False

\frac{44}{320}, \frac{176}{1280}
GCSE Quiz False

\frac{11}{140}, \frac{11}{200}
GCSE Quiz False

Choose two consecutive terms. For example, \frac{11}{20} and \frac{11}{80} .

 

Common ratio,

 

\frac{11}{80} \div \frac{11}{20}=\frac{11}{80} \times \frac{20}{11}=\frac{1}{4}

 

\frac{11}{80} \times \frac{1}{4}=\frac{11}{320}, \frac{11}{320} \times \frac{1}{4}=\frac{11}{1280}

5. Work out the missing terms in the sequence 3, …, …, 24 .

10, 17
GCSE Quiz False

9, 18
GCSE Quiz False

6, 12
GCSE Quiz True

12, 18
GCSE Quiz False

3 has been multiplied by the common ratio, r, three times to get 24.

 

Therefore,

 

3 \times r \times r \times r=24 \text{ or } 3r^{3}=24 .

 

Solving the equation,

 

\begin{aligned} 3r^{3}&=24\\ r^{3}&=8\\ r&=2 \end{aligned}

 

3 \times 2=6, 6 \times 2=12

6. Work out the missing terms in the sequence 90, …, …, \frac{10}{3} .

\frac{550}{9}, \frac{290}{9}
GCSE Quiz False

9, 3
GCSE Quiz False

30, 10
GCSE Quiz True

45, \frac{45}{2}
GCSE Quiz False

90 has been multiplied by the common ratio, r, three times to get \frac{10}{3}.

 

Therefore,

 

3 \times r \times r \times r=24 \text{ or } 3r^{3}=24 .

 

Solving the equation,

 

\begin{aligned} 90r^{3}&=\frac{10}{3}\\ r^{3}&=\frac{10}{3} \div 90\\ r^{3} = \frac{1}{27} r&=\frac{1}{3} \end{aligned}

 

90 \times \frac{1}{3}=30, 30 \times \frac{1}{3}=10

How to generate a geometric sequence

In order to generate a geometric sequence, we need to know the n^{th} term. Using a as the first term of the sequence, r as the common ratio and n to represent the position of the term, the n^{th} term of a geometric sequence is written as ar^{n-1}.

Once we know the first term and the common ratio, we can work out any number of terms in the sequence.

The first term is found when n=1 , the second term when n=2 , the third term when n=3 and so on.

To generate a geometric sequence you need to:

  1. Substitute n=1 into the n^{th} term to calculate the first term.
  2. Substitute n=2 into the n^{th} term to calculate the second term.
  3. Continue to substitute values for n until all the required terms of the sequence are calculated.

How to generate a geometric sequence

How to generate a geometric sequence

Generating a geometric sequence examples

Example 9: generate a geometric sequence using the nth term

Generate the first 5 terms of the sequence 4^{n-1} .

When n = 1,\quad 4^{1-1} = 4^{0} = 1 .

When n = 2,\quad 4^{2-1 }= 4^{1} = 4 .

When n=3, \quad 4^{3-1}=4^{2}=16 .

When n=4, \quad 4^{4-1}=4^{3}=64 .

When n=5, \quad 4^{5-1}=4^{4}=256 .


The first 5 terms of the sequence are 1, 4, 16, 64, 256.

Example 10: generate a geometric sequence using a table

Complete the table for the first 5 terms of the arithmetic sequence 6 \times 2^{n-1}.

n12345
2n-1
6 × 2n-1

n12345
2n-120 = 1
6 × 2n-16

n12345
2n-120 = 121 = 2
6 × 2n-1612

n12345
2n-120 = 121 = 222 = 423 = 824 = 16
6 × 2n-1612244896

Example 11: generate larger terms in a geometric sequence

A geometric sequence has the n^{th} term \left(\frac{1}{2}\right)^{n} .

Calculate the 1^{st}, 2^{nd}, 10^{th} and 12^{th} terms in the sequence. Express your answers as fractions. 

When n=1,\quad \left(\frac{1}{2}\right)^{1}=\frac{1}{2} .

When n=2, \quad \left(\frac{1}{2}\right)^{2}=\frac{1}{4} .

When n=10,\quad \left(\frac{1}{2}\right)^{10}=\frac{1}{1024} .

When n=12,\quad \left(\frac{1}{2}\right)^{12}=\frac{1}{4096} .


The unknown terms are


1, \frac{1}{4}, \frac{1}{1024}, and \frac{1}{4096} .

Example 12: generate a geometric sequence with a negative common ratio

Generate the first 5 terms of the geometric sequence 2(- 3)^{n-1} .

When n=1, \quad 2(-3)^{n-1}=2(-3)^{1-1}=2(-3)^{0}=2 \times 1=2 .

When n=2,\quad 2(-3)^{n-1}=2(-3)^{2-1}=2(-3)^{1}=2 \times-3=-6 .

When n=3, \quad 2(-3)^{n-1}=2(-3)^{3-1}=2(-3)^{2}=2 \times 9=18 .

When n=4, \quad 2(-3)^{n-1}=2(-3)^{4-1}=2(-3)^{3}=2 \times-27=-54 .

When n=5, \quad 2(-3)^{n-1}=2(-3)^{5-1}=2(-3)^{4}=2 \times 81=162 .


The first 5 terms of the sequence are 2, -6, 18, -54, and 162 .

 Geometric sequence practice questions – generate a sequence

1. Generate the first 5 terms of the sequence 10^{n} .

10, 100, 1000, 10000, 100000
GCSE Quiz True

1, 10, 100, 1000, 10000
GCSE Quiz False

10, 20, 30, 40, 50
GCSE Quiz False

100, 1000, 10000, 100000, 1000000
GCSE Quiz False

When  n=1, 10^{1}=10 .

 

When n=2, 10^{2}=100 .

 

When n=3, 10^{3}=1000 .

 

When n=4, 10^{4}=10000 .

 

When n=5, 10^{5}=100000 .

2. Generate the first 5 terms of the sequence 5^{n-1} .

5, 25, 125, 625, 3125
GCSE Quiz False

0.2, 1, 5, 25, 125
GCSE Quiz False

625, 125, 25, 5, 1
GCSE Quiz False

1, 5, 25, 125, 625
GCSE Quiz True

When n=1, 5^{1-1}=5^{0}=1 .

 

When n=2, 5^{2-1}=5^{1}=5 .

 

When n=3, 5^{3-1}=5^{2}=25 .

 

When n=4, 5^{4-1}=5^{3}=125 .

 

When n=5, 5^{5-1}=5^{4}=625 .

3. Generate the first 5 terms of the sequence 4 \times 3^{n-1} .

1, 3, 9, 27, 81
GCSE Quiz False

1, 12, 144, 1728, 20736
GCSE Quiz False

4, 12, 36, 108, 324
GCSE Quiz True

12, 36, 108, 324, 972
GCSE Quiz False

When n=1, 4 \times 3^{1-1}=4 \times 3^{0}=4 .

 

When n=2, 4 \times 3^{2-1}=4 \times 3^{1}=12 .

 

When n=3, 4 \times 3^{3-1}=4 \times 3^{2}=36 .

 

When n=4, 4 \times 3^{4-1}=4 \times 3^{3}=108 .

 

When n=5, 4 \times 3^{5-1}=4 \times 3^{4}=324 .

4. Generate the first 5 terms of the sequence \frac{3^{n}}{6} .

\frac{1}{2}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2}, 40 \frac{1}{2}
GCSE Quiz True

\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}
GCSE Quiz False

\frac{1}{6}, \frac{1}{2}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2}
GCSE Quiz False

\frac{1}{6}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2}, 40 \frac{1}{2}
GCSE Quiz False

When n=1, \frac{3^1}{6}= \frac{1}{2} .

 

When n=2, \frac{3^2}{6}= \frac{9}{6} = 1 \frac{1}{2} .

 

When n=3, \frac{3^3}{6}= \frac{27}{6} = 4 \frac{1}{2} .

 

When n=4, \frac{3^4}{6}= \frac{81}{6} = 13 \frac{1}{2} .

 

When n=5, \frac{3^5}{6}= \frac{243}{6} = 40 \frac{1}{2} .

5. Calculate the 1st, 3rd, 10th and 15th term of the sequence 2^{n}.

2, 4, 8, 16
GCSE Quiz False

2, 8, 1024, 32768
GCSE Quiz True

1, 4, 512, 16384
GCSE Quiz False

4, 16, 2048, 65536
GCSE Quiz False

When n=1, 2^{1}= 2 .

 

When n=3, 2^{3}= 8 .

 

When n=10, 2^{10}= 1024 .

 

When n=15, 2^{15}= 32768 .

6. Calculate the first 5 terms of the sequence 3 \times (-5)^{n-1}.

3, -15, 75, -375, 1875
GCSE Quiz True

1, -5, 25, -125, 625
GCSE Quiz False

1, -225, 3375, -50625, 759375
GCSE Quiz False

-15, 75, -375, 1875, -9375
GCSE Quiz False

When n=1, 3 \times (-5)^{1-1} = 3 \times (-5)^{0} = 3 .

 

When n=2, 3 \times (-5)^{2-1} = 3 \times (-5)^{1} = -15 .

 

When n=3, 3 \times (-5)^{3-1} = 3 \times (-5)^{2} = 75 .

 

When n=4, 3 \times (-5)^{4-1} = 3 \times (-5)^{3} = -375 .

 

When n=5, 3 \times (-5)^{5-1} = 3 \times (-5)^{4} = 1875.

Geometric sequences GCSE exam questions

1. Which sequence is a geometric progression?

 

1, 3, 5, 7, 9,…. \quad \quad \quad 1, 3, 9, 27, 81, …..

 

1, 3, 6, 10, 15, …. \quad \quad 1, 0.6, 0.2, -0.2, -0.6,….

 

(1 mark)

Show answer

1, 3, 9, 27, 81, …..

(1)

2.  Here is a geometric progression,


1, -5, 25, …., 625, …


(a) Find the common ratio.

 

(b) Work out the fourth term of the sequence.

(3 marks)

Show answer

(a)

 

25 \div 5 = – 5

Common ratio = -5

(1)

(b)

 

25 \times -5

(1)

= -125

(1)

3.  A scientist is studying a type of bacteria. The number of bacteria over the first four days is shown below.

 

Day 1 Day 2 Day 3 Day 4
60 180 540 1620

How many bacteria will there be on day 7?

 

(3 marks)

Show answer

180 \div 60 = 3

(1)

1620 \times 3 \times 3 \times 3

(1)

43740

(1)

Common misconceptions

  • Mixing up the common ratio with the common difference for arithmetic sequences

Although these two phrases are similar, each successive term in a geometric sequence of numbers is calculated by multiplying the previous term by a common ratio and not by adding a common difference.

  • A negative value for r means that all terms in the sequence are negative

This is not always the case as when r is raised to an even power, the solution is always positive.

  • The first term in a geometric sequence

The first term is a . With ar^{n-1} , the first term would occur when n = 1 and so the power of r would be equal to 0 . Anything to the power of 0 is equal to 1 , leaving a as the first term in the sequence. This is usually mistaken when a = 1 as it is not clearly noted in the question for example, 2^{n-1} is the same as 1 \times 2^{n-1} .

  • Incorrect simplifying of the nth term

For example, 6 \times 3^{n-1} is incorrectly simplified to 18^{n-1} as 6 \times 3 = 18 .

  • The difference between an arithmetic and a geometric sequence

Arithmetic sequences are formed by adding or subtracting the same number. Geometric sequences are formed by multiplying or dividing the same number.

Learning checklist

You have now learned how to:

  • Recognise geometric sequences

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