Common Number Patterns

Numbers can have interesting patterns.
Here we list the most common patterns and how they are made.

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Arithmetic Sequences

An Arithmetic Sequence is made by adding the same value each time.

Example:
1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:

arithmetic sequence 1, 4, 7, 10,

Example:
3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:

arithmetic sequence 3, 8, 13, 18

The value added each time is called the "common difference"

What is the common difference in this example?

19, 27, 35, 43, ...

Answer: The common difference is 8

The common difference could also be negative:

Example:
25, 23, 21, 19, 17, 15, ...

This common difference is −2
The pattern is continued by subtracting 2 each time, like this:

arithmetic sequence 25, 23, 21,...

1739, 1740, 2511, 9771, 9772

 

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Geometric Sequences

A Geometric Sequence is made by multiplying by the same value each time.

Example:
1, 3, 9, 27, 81,243, ...

This sequence has a factor of 3 between each number.
The pattern is continued by multiplying by 3 each time, like this:

geometric sequence 1, 3, 9,

What we multiply by each time is called the "common ratio".

In the previous example the common ratio was 3:

geometric sequence 1, 3, 9, common ratio 3

We can start with any number:

Example: Common Ratio of 3, But Starting at 2:

2, 6, 18, 54,162,486, ...

This sequence also has a common ratio of 3, but it starts with 2.

geometric sequence 2, 6, 18

Example:
1, 2, 4, 8, 16, 32, 64,128,256, ...

This sequence starts at 1 and has a common ratio of 2.
The pattern is continued by multiplying by 2 each time, like this:

geometric sequence 1, 2, 4, 8, 16,

The common ratio can be less than 1:

Example:
10, 5, 2.5, 1.25, 0.625, 0.3125, ...

This sequence starts at 10 and has a common ratio of 0.5 (a half).
The pattern is continued by multiplying by 0.5 each time.

But the common ratio can't be 0, as we get a sequence like 1, 0, 0, 0, 0, 0, 0, ...

658, 796, 1741, 10006, 10007, 3861, 3862, 8283, 8284, 3860

Special Sequences

There are also many special sequences, here are some of the most common:

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This Triangular Number Sequence is generated from a pattern of dots that form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

triangular numbers

Square Numbers

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...

They are the squares of whole numbers:

0 (=0×0)
1 (=1×1)
4 (=2×2)
9 (=3×3)
16 (=4×4)
etc...

Cube Numbers

1, 8, 27, 64,125,216,343,512,729, ...

They are the cubes of the counting numbers (they start at 1):

1 (=1×1×1)
8 (=2×2×2)
27 (=3×3×3)
64 (=4×4×4)
etc...

Fibonacci Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The Fibonacci Sequence is found by adding the two numbers before it together.
The 2 is found by adding the two numbers before it (1+1)
The 21 is found by adding the two numbers before it (8+13)
The next number in the sequence above would be 55 (21+34)

Can you figure out the next few numbers?

Other Sequences

There are lots more! You might even think of your own ...

 

1736, 26140, 26145, 1735, 26133, 15008, 26151, 26152, 1737, 1738