In Fig. \(\)5, we have the first fourteen Fibonacci numbers in the chart. We can see multiples of numbers begin to evolve in this chart and the multiples continue on. By this, when \(n=3,\) the Fibonacci number is \(x_n=2\) and when \(n=6,\ \) \(x_n=8.\)
- From here, we can conclude that every third Fibonacci number is a multiple of two.
In addition, this pattern holds for other multiples and is true when \(n=4,\ x_n=3\) and when \(n=8,\ x_n=21.\)
- This means that every fourth Fibonacci number is a multiple of three.
We can see that this pattern will continue on since we see that every \(5^{th}\) Fibonacci number is a multiple of five.
Another pattern seen within Fig. 5 is that the Fibonacci numbers go in a pattern that is (odd, odd, even). This is one of Fibonacci's theorems that says: For any positive integer \(m\), \(F_{3m-2}\) and \(F_{3m-1}\) are odd numbers and \(F_{3m}\) are even numbers.
By this, any natural number is of the form \(3m-2,\ 3m-1\) and \(3m\) and using a proof by induction will show the result for the Fibonacci sequence.