## Number Sequences

Number sequences are simply a group of numbers arranged so that they follow a specific pattern or mathematical rule. For example, see if you can determine the next three numbers in each of these number sequences.

2, 7, 12, ...

3, 6, 9, 12, ...

5, 2, -1, -4, ...

1, 1.5, 2, 2.5, 3, ...

Pretty simple so far, right? Now try these.

3, 6, 12, 24, ...

4, -8, 16, -32, ...

0.25, 0.5, 0.75, 1, ...

Now for a bit more of a challenge...

-1, 0, 3, 8, ...

0.5, 2, 4.5, 8, ...

As you may have noticed, figuring out the next number in the sequence is all about understanding the

One of the most famous mathematical sequences is known as the Fibonacci sequence, named after the Italian mathematician Leonardo Pisano Bigollo. He was commonly known a Fibonacci, and lived in Pisa during the early 13th Century. He is not only known for this famous sequence of numbers, but he revolutionized mathematics in Medieval Europe, almost singlehandedly. He's actually quite a fascinating guy; you can read more about him here.

Here are the first five terms of the Fibonacci sequence: 1, 1, 2, 3, 5, ...

Can you find the rule, and figure out the next three numbers in the sequence?

Now see if you can find the 16th number in the sequence.

Then, watch this video of a song called

A couple of things about the video before you start watching it. The caption at around 1:25 into the video says, "The time signature of this main riff is 9-8-7", which is not really a time signature. What it means to say is that the time signature changes from 9/8 to 8/8 (the same as 4/4), and then to 7/8, and repeats this sequence three times (3 being a number in the Fibonacci sequence.) Even if you're not a musician, see if you can recognize this sequence of time changes in the music. The "riff" comes in the first time at 1:13, and it happens a couple of more times later in the song, during each chorus. Fair warning: it's a bit tricky to figure out. Also, when the vocal part comes in at 1:38, the time signature is 5/8, 5 also being a Fibonacci number.

As you can see, number sequences can appear in the most unexpected places...

2, 7, 12, ...

3, 6, 9, 12, ...

5, 2, -1, -4, ...

1, 1.5, 2, 2.5, 3, ...

Pretty simple so far, right? Now try these.

3, 6, 12, 24, ...

4, -8, 16, -32, ...

0.25, 0.5, 0.75, 1, ...

Now for a bit more of a challenge...

-1, 0, 3, 8, ...

0.5, 2, 4.5, 8, ...

As you may have noticed, figuring out the next number in the sequence is all about understanding the

*pattern*or*mathematical rule*that defines the sequence.One of the most famous mathematical sequences is known as the Fibonacci sequence, named after the Italian mathematician Leonardo Pisano Bigollo. He was commonly known a Fibonacci, and lived in Pisa during the early 13th Century. He is not only known for this famous sequence of numbers, but he revolutionized mathematics in Medieval Europe, almost singlehandedly. He's actually quite a fascinating guy; you can read more about him here.

Here are the first five terms of the Fibonacci sequence: 1, 1, 2, 3, 5, ...

Can you find the rule, and figure out the next three numbers in the sequence?

Now see if you can find the 16th number in the sequence.

Then, watch this video of a song called

*Lateralus,*by the band Tool, which is based on the Fibonacci sequence, and specifically the 16th term of the sequence.A couple of things about the video before you start watching it. The caption at around 1:25 into the video says, "The time signature of this main riff is 9-8-7", which is not really a time signature. What it means to say is that the time signature changes from 9/8 to 8/8 (the same as 4/4), and then to 7/8, and repeats this sequence three times (3 being a number in the Fibonacci sequence.) Even if you're not a musician, see if you can recognize this sequence of time changes in the music. The "riff" comes in the first time at 1:13, and it happens a couple of more times later in the song, during each chorus. Fair warning: it's a bit tricky to figure out. Also, when the vocal part comes in at 1:38, the time signature is 5/8, 5 also being a Fibonacci number.

As you can see, number sequences can appear in the most unexpected places...

After you have finished this learning activity, continue to the page "Identifying and Using Arithmetic Sequences", or click on the button below.