Other sequences - KS3 Maths

Key points

A sequence with the terms two, five, ten, seventeen, twenty six. Written above: between each pair of terms is the amount the sequence is increasing by. Plus three, plus five, plus seven, plus nine, with curved arrows going from left to right coloured blue. — Image caption,
Non-linear sequences increase or decrease by different amounts each time.

A number pattern is a series of numbers that follow a specific rule. Examples include: odd numbers, even numbers, square numbers and multiples.

A good knowledge of arithmetic and geometric sequences is essential to understanding other types of .

Each number in a sequence is called a term.
A sequence which increases or decreases by the same amount each time is called a .
The of a sequence describes how to get from one term to the next.
There are other types of sequence that follow a rule, but increase or decrease by a different amount each time.

Finding missing terms in a non-linear sequence

Linear sequences increase or decrease by the same amount each time.

Not all sequences are linear. The term-to-term differences may not be the same and the next term is not always obvious.

There is often a pattern between the differences, rather than the terms, which helps to predict the next number in the sequence.

Example

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What is the next term in the sequence?
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The first step is to look at the difference between each pair of numbers in the sequence to find the term-to-term rule. The difference between each pair of terms is a different amount each time. 3 – 1 = 2, 6 - 3 = 3, 10 – 6 = 4 and 15 – 10 = 5
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While the sequence is not linear, there is a pattern between the difference of each pair of terms. If the pattern continues, then the next difference will be +6, meaning that the next term is 21 (15 + 6).
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Some sequences have special names. This is a sequence of triangular numbers. The name can be understood by looking at diagrams of triangles made of dots. The triangle has one dot on the first row, two on the second row, three on the third row, and four on the fourth row, and so on.
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The number of dots arranged in a triangular formation is the same as the number in the sequence. This creates the name triangular numbers.
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The sixth diagram in the sequence will use the existing 15 dots and place 6 more dots underneath to create a bigger triangle. Now there are 21 dots required in total to make the triangle.
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The triangular formation can be arranged in different ways but still use the same number of dots each time.
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Looking at the difference between consecutive terms can also reveal the pattern. The difference between each term is increasing in ones. 3 – 1 = 2, 6 – 3 = 3, 10 – 6 = 4, and 15 – 10 = 5. The next difference will be 6. Therefore, the next number in the sequence is 21 (15 + 6).

Question

What is the next term in this sequence?

A sequence with the terms three, five, eight, twelve.

Finding the next term in a quadratic sequence

involve .

They can be identified by the fact that the differences between the terms are not equal.

However, the difference between the differences (known as the second difference) is equal.

The term-to-term rule and the second difference can be used to find the next number in a quadratic sequence.

Examples

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What is the next term in this sequence?
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The first difference between the terms is not the same each time. However, there is a pattern between the first differences. Each difference increases by 2
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+3, +5, +7, +9 are the first differences. +2, +2, +2 are the second differences. The second difference is 2. This means that the sequence is quadratic.
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Continue this pattern of differences to find the next number in the sequence.
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The first difference will increase by 2 each time. The difference between the fifth and the sixth term will be +11. The next term of the sequence is 26 + 11 = 37. The sequence involves square numbers as each term is one more than a square number.
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What is the next term in this sequence?
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The first difference between the terms is not the same each time. However, there is a pattern between the first differences. Each difference increases by 2 each time.
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The second difference is +2. 9 – 7 = 2, 11 – 9 = 2 and 13- 11 = 2. +7, +9, +11, +13 are the first differences. +2, +2, +2 are the second differences. The second difference is 2. This means that the sequence is quadratic.
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The difference between the fifth and the sixth term will be +15. Continue this pattern of differences to find the next term in the sequence. The sixth number in the sequence is 59, because 44 + 15 = 59

Question

What is the next term in this sequence?

A sequence with the terms four, seven, twelve, nineteen, twenty eight.

Practise other sequences

Quiz

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

Real-life maths

An image of a sunflower with a Fibonnaci spiral overlaid. — Image caption,
The Fibonacci sequence can be pictured as a spiral, like the pattern of sunflower seeds.

Non-linear sequences can be seen in nature.

The number of petals on many flowers and the branches of trees grow in a ‘Fibonacci sequence’.

This is a naturally occurring pattern where the previous two numbers are added together to create the next number in the sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34…).

Some flowers have 3 petals, others have 5 petals, others 8 petals, and so on.

Fibonacci numbers can be pictured as a spiral, which can also be seen in hurricanes, seashells and the pattern of sunflower seeds.