# Sequence pattern 3 - multiples

The next pattern to look for in a sequence are multiples.

They also call this a geometric sequence or geometric progression.

`3`, `6`, `12`, `24`, `48`

Divide each term by the previous term

`6/3=2`, `12/6=2`, `24/12=2`, `48/24=2`

The multiple used is 2.

**Example 1**

What are the next two numbers in the following sequence?

`2`, `6`, `18`, `54`, `162`

Is it arithmetic - a consistent difference between numbers?

No it's not

Is it quadratic - a consistent difference between differences?

No it's not

Is it a multiple (or geometric sequence)?

`2`, `6`, `18`, `54`, `162`

Divide each term by the previous term

`6/2=3`, `18/6=3`, `54/18=3`, `162/54=3`

Yes, it is. It is a multiple of 3.

So the next two numbers are:

`162times3=486`

and the next would be

`486times3=1,458`

**Example 2**

What are the next two numbers in the following sequence?

`-2`, `4`, `-8`, `16`, `-32`, `64`

Is it arithmetic - a consistent difference between numbers?

No it's not

Is it a quadratic - a consistent difference between differences?

No it's not

Is it a multiple (or geometric sequence)?

`-2`, `4`, `-8`, `16`, `-32`, `64`

Divide each term by the previous term

`4/-2=-2`, `-8/4=-2`, `16/-8=-2`, `-32/16=-2`, `64/-32=-2`

Yes, it is. It is a multiple of `-2`.

So the next two numbers are:

`64times-2=-128`

and the next would be

`-128times-2=256`

Answer the next two numbers are `-128` and `256`

**Example 3**

What are the next two numbers in the following sequence?

`40`, `20`, `10`, `5`, `2.5`

Is it arithmetic - a consistent difference between numbers?

No it's not

Is it quadratic - a consistent difference between differences?

No it's not

Is it a multiple (or geometric sequence)?

`40`, `20`, `10`, `5`, `2.5`

Divide each term by the previous term

`20/40=0.5`, `10/20=0.5`, `5/10=0.5`, `2.5/5=0.5`

Yes, it is. It is a multiple of `0.5`.

So the next two numbers are:

`2.5times0.5=1.25`

and the next would be

`1.25times0.5=0.625`

Answer the next two numbers are `1.25` and `0.625`

**Example 4**

What are the next two numbers in the following sequence?

`3`, `3sqrt8`, `24`, `24sqrt8`

This is a trick because the only way to tackle this is to:

- Try and keep the first two sequences going, which is

`3` and `3sqrt8` - Then you have to recognise that

`8=sqrt8timessqrt8`

So this sequence could be rewritten

`3` | `=3` | ||

`3sqrt8` | `=3sqrt8` | ||

`24` | `=` | `3timessqrt8timessqrt8` (i.e. `3times8=24`) | `=3sqrt8sqrt8` |

`24sqrt8` | `=` | `(3timessqrt8timessqrt8)timessqrt8` | `=3sqrt8sqrt8sqrt8` |

So the sequence can be rewritten as:

`3`, `3sqrt8`, `3sqrt8sqrt8`, `3sqrt8sqrt8sqrt8`

Now you can clearly see that dividing each term by the previous term gives

`(3sqrt8)/3=sqrt8`, `(3sqrt8sqrt8)/(3sqrt8)=sqrt8`, `(3sqrt8sqrt8sqrt8)/(3sqrt8sqrt8)=sqrt8`

So this is a multiple of `sqrt8`

The next two numbers in the sequence are

`3sqrt8sqrt8sqrt8timessqrt8=3times8times8=192`

and the next is

`3sqrt8sqrt8sqrt8sqrt8timessqrt8=3times8times8timessqrt8=192sqrt8`

Answer next two numbers are `192` and `192sqrt8`