Finding number patterns in arithmetic sequences

Part ofMathsPatterns and sequences

Key points

A sequence with the terms nineteen, fifteen, eleven, seven. Written above: between each pair of terms is the amount the sequence is decreasing by. Subtract four, subtract four, subtract four, with curved arrows going from left to right coloured blue. Written right: Subtract four, with a curved arrow going from left to right coloured orange. The next term, three, has been added to the end of the sequence written in orange.
An example of a linear sequence.

  • A list of numbers or diagrams that are in a particular order is called a

  • A which increases (or decreases) by the same amount each time is called a linear sequence.

  • The amount it increases or decreases by is known as the .

  • Recognising the common difference means that the sequence can be continued using a .

A sequence with the terms nineteen, fifteen, eleven, seven. Written above: between each pair of terms is the amount the sequence is decreasing by. Subtract four, subtract four, subtract four, with curved arrows going from left to right coloured blue. Written right: Subtract four, with a curved arrow going from left to right coloured orange. The next term, three, has been added to the end of the sequence written in orange.
An example of a linear sequence.
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Finding the next term in an arithmetic sequence

  • The first step is to find the common difference between in the arithmetic sequence.
  • If the arithmetic sequence is increasing, then the common difference will be to add the same amount each time.
  • An arithmetic sequence can also decrease, which means that the common difference will be to subtract the same amount each time.
  • Add (or subtract) the common difference to the last given number in the sequence.

Examples

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  1. Example one. A sequence with the terms two, five, eight, eleven.

    What is the next number in the sequence?

Question

What is the next number in this arithmetic sequence?

A sequence with the terms thirty one, twenty six, twenty one, sixteen.

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Number patterns from diagrams

  • Sequences can be a series of .
  • A number pattern in a diagram often requires counting shapes to find the rule.
  • Look at how the pattern grows from one term to the next.
  • Sometimes there are different elements (or shapes) in the diagram that decrease or increase in different ways.
  • Knowing how the shapes decrease or increase makes it possible to draw the next pattern in the sequence.

Examples

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  1. A sequence made of three images. The first image has two squares, joined side by side. A circle sits on top of the left square. The second image has two rows of two squares, joined side by side. Two circles sit vertically above the top left square. The third images has three rows of two squares, joined sided by side. Three circles sit vertically above the top left square. Each image is labelled, pattern one, pattern two, pattern three. The squares are coloured red and the circles are yellow..

    How many yellow circles and red squares will be needed to make the next pattern in the sequence?

Question

How many red tiles will be needed to make the fourth pattern?

A sequence made of three images. The first image has three red squares, joined side by side. The second image is the same as the first image with an additional row of three squares below. These are coloured red, blue, red. The third image is the same as the second image with an additional row of three squares below. These are also coloured red, blue, red. Each image is labelled, pattern one, pattern two, pattern three.

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Practise linear arithmetic sequences

Quiz

Practise exploring number patterns in linear sequences with this quiz. You may need a pen and paper to help you with your answers.

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Real-life maths

Two images showing distributions of tables and chairs. The first image has three red squares side by side with three blue circles above and below. The second image is identical to first image with the addition of one blue circle at either end of the joined red squares.

Using sequences can be helpful in everyday life.

Hospitality venues that host events can offer a variety of ways in how they set up a room. They can organise how tables and chairs should be arranged, depending on how many people are attending an event and the layout of the room.

For example, one option could be to have four guests seated around a table (represented in the image by one red block) or instead to have six guests seated around a table (represented by two red blocks).

Using the same seating arrangements, the event organiser can use similar sequences to calculate how many guests can then be seated at three tables, or more.

Two images showing distributions of tables and chairs. The first image has three red squares side by side with three blue circles above and below. The second image is identical to first image with the addition of one blue circle at either end of the joined red squares.
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