Is there a pattern to the golden ratio number figures? The golden ratio or phi is 1.6180339887498948482045... I am wondering if there is a pattern in the numbers so given a certain set of figures, you are able to figure out the rest of the figures aaccurately? Essentially is there a pattern that someone could follow and say you were given 1.6180339887 and from the pattern formulated you can figure that the next numbers are 498948...?
 A: Given a finite sequence of numbers, it is impossible to answer the question what is the next number of the sequence.  Such questions usually rely on there being a fairly simple recognizable pattern, but there is always an implicit assumption that the pattern will continue.  However, there is no guarantee that this is true.  
Suppose that a sequence starts: $1,4,7$.  If asked for the next number of the sequence, most people would respond $10$.  However, if you pick any positive integer $n$, there is a degree 3 polynomial $P$ such that $P(1)=1, P(2)=4, P(3)=7$ and $P(4)=n$.  So your question is ill conceived.
A: You can calculate all of the digits of $\phi$ without having any of the previous digits by just evaluating $\frac{1+\sqrt{5}}{2}$. However, if you are trying to find a pattern in the digits, this is an unsolved problem. If the digits of a number are evenly distributed and every digit is just as likely as any other digit, then the number is called normal. Proving that a number is normal is extremely difficult unless the number was defined to have uniformly distributed digits. You can read more here.
A: The Golden ratio is well-known for being written with repeating patterns, such as :  $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}$$
or $$\phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}\,.$$
It can be written as a series:
$$\phi=\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}\,.$$
It can be derived from the Fibonacci sequence, where several patterns exist. Yet, I am not unware of a pattern in its digits in base $10$.
