# Common Number Patterns

### Numbers can have interesting patterns.

Here we list the most common patterns and how they are made.

## 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:

### 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:

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:

## 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:

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

In the previous example the common ratio was **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**.

### 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:

The common ratio can be less than 1:

### Example:

10, 5, 2.5, 1.25, 16, 32, 64, 128, 256, ... |

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 would get a sequence like 1, 0, 0, 0, ...

## Special Sequences

**Triangular Numbers**

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

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

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

## 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 ...