One of the interesting things about the Fibonacci sequence is the fact that is converges on the golden ratio. For starters, let us merely observe that it, in fact, converges:
If we take the table of n and F(n), We can lay them out as fractions:
If we find the decimal values for these fractions, we notice that they converge. This graph shows the convergence:
We can define each successive xn as such:
But this does not show us exactly what it converges to. To find this, we will start out by writing the recursive definition for the Fibonacci sequence.
We know that the Fibonacci sequence converges, so we know that their limits are the same.
This tells us, then, that the following must be true:
With this reduced form, it is very easy to solve using the quadratic equation. x is found to be:
We can use this to find the nth Fibonacci number. If we set phi to the value we discovered for x, we can do some interesting things. We know that (phi) + (phi - 1) = 2phi-1, or the square root of 5. By dividing both sides by the square root of 5, we get get this equation, which is equal to 1:
By raising phi and phi-1 to various powers, we always get integers, and interestingly enough, when we raise it to the power of n-1, we get the nth Fibonacci number:
(Graph courtesy of Dr. Ron Knott, FIMA, C.Math, MBcs,C.Eng, Dept. of Mathematical and Computing Sciences, Univerity of Surrey, UK
Other graphics contained on this page courtesy of MathAcademy)