Fibonacci and other famous number patterns

As a student of mathematics, you’re bound to come across the Fibonacci sequence. It’s a well-known number pattern that can be applied to a wide range of subjects, from music and art to finance and nature. And there are more of these number patterns that are used to help us make sense of the world.
 
Decoding the Fibonacci Sequence
 
Take a look at the following series of numbers and see if you can figure out the pattern:
 
0, 1, 1, 2, 3, 5, 8, 13, 21, 34…
 
Would you be able to work out what the next number in the sequence is? If you answered 55, you’re correct – and it means you’ve cracked the code of the famous Fibonacci sequence.  It’s quite simple to solve as each number is simply the sum of the two numbers that come before it.
 
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
 
Now you might be wondering who the first person to notice this pattern was, and how they discovered it.  The answer is a 12^th century Italian mathematician known as Leonardo Pisano who wrote a book called Liber Abaci or ‘The Book of Calculations’.
 
It All Started with Rabbits
 
In his book, Leonardo Pisano posed a mathematical problem that involved working out how quickly a pair of rabbits could breed. If we imagine that one male and one female rabbit would take one month to produce another pair of rabbits (and assuming that the rabbits live forever), how many pairs of rabbits would there be at the end of a year?
 
The formula he came up with to solve this problem is:

X_n + 1 = X_n + X_n - 1
 
And you can use this same formula to generate Fibonacci numbers. In case you were wondering, after one year there would be 233 pairs of rabbits.
 
Patterns in Nature
 
You may be surprised to learn that the Fibonacci Sequence can be observed in the natural world, especially in trees and flowers. For example, in the way the seeds of a sunflower are arranged. Look closely and you will find alternating rows of seed spiralling outward from the centre, one row consisting of 21 seeds, the next of 34 seeds and then another 55 seeds. It appears that this is most efficient way to pack all the seeds uniformly – no matter how big the flower grows.
 
Even pine cones and several species of trees demonstrate the pattern that is referred to as the Fibonacci spiral. This spiral can be created by arranging squares with sizes that match the numbers in the Fibonacci sequence and drawing a quarter of a circle in each square.


 
More Amazing Number Sequences
While the Fibonacci Sequence is one of the most fascinating number patterns, there are many more such as:
 
Arithmetic Sequence
In this case, the pattern is the result of a rule that’s applied consistently, such as adding or subtracting the same number each time. For example in this sequence, the common difference between the numbers is 3: 1, 4, 7, 10.
 
Geometric Sequence
Here, a list of numbers is generated by multiplying or dividing them by the same amount. As an example, look at this pattern and you’ll see that each number has been multiplied by the number 2: 2, 4, 8, 16, 32.
 
Square and Cube Sequence
These patterns are generated by calculating the squares of whole numbers and the cubes of counting numbers, as follows:
 


Why Study Number Patterns
 
Now that you know more about number patterns, you might try making up your own.
 
How are number patterns useful?
Aside from being fascinating, studying number patterns an important purpose. As a maths student, being able to recognise and work out a pattern can help you solve problems. Often, you’ll be able to use a simple pattern to breakdown and solve a more complex problem.
 
Patterns are also useful for predicting future events and identifying and applying patterns is a part of developing logical reasoning abilities.
 
The subject of mathematics is full of unusual phenomena like number patterns. To help maths students from Grade 1 right through to Grade 12 develop their skills, ABC Maths and Science created a range of electronic exercises and worksheets to help everyone master maths. For more information on what we offer, please send an email to info@abcbooks.co.za.


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