What is a geometric sequence?

  • A geometric sequence goes from one term to the next by always multiplying or dividing by the same value.

  • The number multiplied (or divided) at each stage of a geometric sequence is called the common ratio.

  • Examples of geometric sequences are the frequencies of musical notes and interest paid by a bank.

Find out how notes on a keyboard relate to geometric sequences.

What is a geometric sequence?

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.

The sound of a geometric sequence

Geometric sequences are important in music. Musical notes each have a frequency measured in Hertz (Hz). The higher the note, the higher the number of Hertz.

For example, the note A can be played with a frequency of 110 Hz, 220 Hz, 440 Hz, 880 Hz...

Each term has been multiplied by the same number to make the next term in the sequence.

Each of the specific frequencies is called a pitch. The interval between the pitches is called an octave.

The common ratio

The common ratio is the number you multiply or divide by at each stage of the sequence.

You can find it by dividing two consecutive pairs of terms. It doesn't matter which pair is chosen as long as they are next to each other:

220 ÷ 110 = 2

440 ÷ 220 = 2

880 ÷ 440 = 2

The common ratio is therefore 2. You can find out the next term in the sequence by multiplying the last term by 2.

A visual representation of the geometric sequence described in the article.
Here is the sequence written out with the common ratio.

Where next?

Discover more about this topic from around Bitesize.

Revise - Linear sequences
How to visualise an arithmetic sequence
What is a number line?
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