The Fibonacci numbers are infinite sequence of numbers named after the Italian mathematician Leonardo di Pisa, later known as Fibonacci. He introduced this sequence of numbers in one of his books in 1202. The sequence is

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,……..

The way this sequence works is that every number in the sequence is the sum of the two numbers before it:

·       2 is found by adding the two numbers before it (1+1)

·       3 is found by adding the two numbers before it (1+2)

·       5 is (2+3)

·       and so on!

Below are some fun facts about this sequence of numbers:

1-    Final Digits

Let's look at the final digit in the Fibonacci sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,……..

The sequence of final digits is

1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7,…..

Is there a pattern in this new sequence?

Yes!

If you write many terms of this new final digit sequence, you will notice that this sequence repeats the same sequence after each 60 terms. We say the series of digits repeats with a cycle of length 60.

If you similarly make a sequence consisting of the last two digits of the Fibonacci sequence, it turns out that this new sequence has a cycle of length 300. If you use the last three digits, the cycle has length 1500.

2-    Factors of Fibonacci numbers

The 3rd Fibonacci number is 2 so every 3rd number in the sequence is a multiple of 2 (2, 8, 34, 144, 610,…)

The 4th Fibonacci number is 3 so every 4th number in the sequence is a multiple of 3 (3, 21, 144,….)

The 5th Fibonacci number is 5 so every 5th number in the sequence is a multiple of 5 (5, 55, 610,….)

The general rule is that if n is the kth Fibonacci number, then every kth number in the sequence will be a multiple of n.

3-    Golden Ratio

Let us try dividing each Fibonacci number beginning from 3 by the number in the sequence that proceeds it.

3/2=1.5

5/3=1.6666666….

8/4=1.6

13/8=1.625

If we continue making these divisions, we notice that the ratios get closer to the number 1.6180339887…. This number is called the golden ratio ϕ

 A golden rectangle is a rectangle whose length and width have ratio ϕ. A series of interlocked golden rectangles creates the shape of the golden spiral as seen below