Some Famous Number Sequences

 

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Number sequences

Like numbers, number sequences can become famous too. Most of the number sequences are famous because they are very simple, but exhibit some very interesting properties. Number sequences are like pancakes - the simpler they are the better - the art is in the filling!

Two sequences that you already know about are sequences of even and odd numbers. But there is one number sequence that is more famous than any other, and this is the one we will explore now - it is called Fibonacci Sequence after a mathematician that invented it.

He came up with the idea when he thought about how rabbits breed (there is a lesson to be learnt here - especially if you plan visiting the pet shop soon!). You can work that out for yourself by downloading the worksheet on Fibonacci rabbits.

Fibonacci's number sequence looks like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on...

Can you work out the pattern? Can you describe it?

It is pretty simple really - you add the last two terms of your number sequence to get the new term. Now... let's just look at what a term means - in this Fibonacci's sequence this is how the terms will be named:

1 st term

2 nd term

3 rd term

4 th term

5 th term

6 th term

7 th term

8 th term

9 th term

10 th term

and so on...

n - th term

1

1

2

3

5

8

13

21

34

55

89

?

Can you work out the 50 th term for Fibonacci's sequence?

In fact this type of sequence doesn't have to start with 1. It can start with any number, as long as the principle employed is the same. Download worksheet no. 2 on Fibonacci's sequence to try and generate some other sequence of the same kind. Part of this worksheet will require you to have access to a spreadsheet programme, such as Excel.

A very interesting fact about Fibonacci's numbers is that when you divide one of the numbers from the sequence with the preceding number, you get approximately the same number, which is called . You can click on it to learn more about the number itself.

This number turns up in all kinds of places and in all kinds of ways. It's value is

Try to calculate its approximate value on your calculator. Remember, is an irrational number, which means that you can not find its precise value on the calculator (or in any other way!).

If you use this number and the Fibonacci number sequence to draw geometrical figures, you may end up constructing various figures: Golden Ratio (or Golden Mean), pentagon (using the Golden Mean), logarithmic spiral, which shows the principles for the growth of some plants and animals such as Nautilus and sunflower. There is yet another worksheet on this! Download it here and enjoy!

Golden Rectangle is divided in such a way that the length of the side of the larger square is equal to the lenght of the sum of the lengths of the sides of two preceding smaller squares... or something like that! Download the worksheet to try for yourself.

You can use the Golden Rectangle to draw this beautiful spiral.

 

   

Fibonacci's Number Sequence ...

got the name after the mathematician who invented it: you guessed it - his name was Fibonacci. Click on his portrait below to learn more about him.

Fibonacci's Number Sequence can be spotted in nature - spirals in these flowers and in the shell follow the patterns of Fibonacci's numbers.

just one of the illustrations from nature - cactus flower

sunflower seeds...

 

shell named Nautilus...

Download some worksheets to investigate Fibonacci's numbers further by clicking on the Number Man

 

 

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