What are geometric sequences?

Key points

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Each term in this sequence is doubling (multiplying by 2) to create the next term.

A is a list of numbers or diagrams that are in order.

Number sequences are sets of numbers that follow a pattern or a rule.
If the rule is to multiply or divide by a specific number each time, it is called a geometric sequence.
A number pattern which increases (or decreases) by the same amount each time is called an .
Recognising the pattern between the means that the sequence can be continued using a rule.
Sequences that are connected by relationships are geometric sequences, whereas those connected by relationships are linear.

Finding missing terms in a geometric sequence

Each term in a geometric sequence is found by multiplying or dividing the previous term by the same amount, this is called the common ratio.
To find the common ratio, start by calculating the difference between each pair of numbers, moving from one term to the next.

Examples

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What is the next term in the sequence?
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To work out the next term in the sequence, look at the difference between each pair of numbers. The difference between each pair of terms is a different amount each time. The difference between 1 and 2 is 1, the difference between 2 and 4 is 2, the difference between 4 and 8 is 4 and so on. The number pattern does not increase by the same amount each time. It is not an arithmetic linear sequence.
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Although the sequence is not arithmetic, there is a pattern between the difference of each pair of terms. If the pattern continues, the next difference will be +16. This means that the next term is 32
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Each term in the sequence is doubling (multiplying by 2) to create the next term. This is a multiplicative relationship. The term-to-term rule is ×2
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As each term is multiplied (or divided) by the same number (2) to make the following term, this sequence is called a geometric sequence. The next term in the sequence will be 32 (16 x 2).
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What is the next number in this sequence?
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The common difference between a pair of terms is found by subtracting the smaller term from the larger term. 6 – 2 = 4, 18 – 6 = 12 etc. This shows that the sequence is not a linear arithmetic sequence.
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There is a multiplicative relationship between each pair of terms. Each term in the sequence is multiplied by 3 to create the following term. The term-to-term rule is ×3. The sequence will continue in the same way, multiplying the previous term (54) by 3 will give the next term.
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The next term will be 162 (54 x 3).

Question

What is the next number in this geometric sequence?

A sequence with the terms eighty, forty, twenty, ten.

Finding the common ratio

The common ratio is the number you multiply or divide by at each stage of the sequence. It is found by dividing two consecutive pairs of terms.
It does not matter which pair of terms is chosen, as long as they are next to each other in the sequence.

Examples

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What is the common ratio of this geometric sequence of numbers?
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100 multiplied by the common ratio gives the next term, 200. To calculate the common ratio, calculate 200 ÷ 100 = 2. This means that the common ratio is 2. 100 × 2 = 200, 200 × 2 = 400, 400 × 2 = 800
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What is the common ratio of this geometric sequence of numbers?
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The common ratio is found by dividing two consecutive pairs of terms. The first 2 terms in the sequence are 50 and 200. 200 ÷ 50 = 4, 800 ÷ 200 = 4 and 3200 ÷ 800 = 4. The common ratio is 4. The next term in the sequence is found by multiplying the last term by 4
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What is the common ratio of this geometric sequence of numbers?
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The common ratio is found by dividing two consecutive pairs of terms. The first 2 terms in the sequence are 4000 and 2000. 2000 ÷ 4000 = 0∙5, 1000 ÷ 2000 = 0∙5 and 500 ÷ 1000 = 0∙5. The common ratio is 0∙5. The next term in the sequence is found by multiplying the last term by 0∙5
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Multiplying the last term by 0∙5 will produce the next term. This relationship could be described using division as well as multiplication. Each term has also been divided by 2 to reveal the next term. 4000 ÷ 2000 = 2, 2000 ÷ 1000 = 2 and 1000 ÷ 500 = 2. The next term in the sequence is found by dividing the previous term by 2
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Multiplying by 0∙5 and dividing by 2 produces the same answer. The next term in the sequence can be found by working out 500 x 0∙5 or 500 ÷ 2. Both reveal that the next term is 250

Question

What is the common ratio of this geometric sequence?

A sequence with the terms one, ten, one hundred, one thousand.

Practise geometric sequences

Quiz

Practise recognising and finding terms in geometric sequences with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of a microbiologist inspecting a petri dish. — Image caption,
Scientists like microbiologists use geometric sequences to monitor the growth of bacteria.

Geometric sequences are used in everyday life when something follows a pattern. An example of this is in scientific work, when the growth of bacteria is monitored.

If provided with the optimum conditions for growth, the population of bacteria grows in a geometric sequence, because bacteria reproduce by dividing into two.

Scientists, such as microbiologists, can predict how much bacteria will develop in a petri dish after a certain number of days by finding the common ratio.