The golden mean or golden ratio is a very important number for its occurrences in mathematics, other sciences and nature. It is intrinsically related with the Fibonacci numbers. The golden mean is the positive root of the polynomial x2-x-1, i.e. if Φ denotes the golden ratio, then it satisfies the equation Φ2=Φ+1 and Φ>0. In the present post I will show that it satisfies this beautiful formula:
The exact value of the golden mean is
It is quite easy to prove the identity (1). If x satisfies the equation x2=x+1, taking in account that x is positive, then it holds
If we substitute the variable x into the previous equation we have
Repeating again this process, it follows
After n-times, the following identity holds
Therefore in the limit, we have the identity (1). Moreover, it can be used to approximate the value of Φ, as follows: Let
The limit of xn, when n goes to infinity, is Φ. It can be proved that for any choice of x1 >-1, the corresponding sequence xn has the same limit Φ.
On the other hand, we recall that the equation x2-x-1 has other root , a negative number, i.e.
We can have an identity similar to (1) for τ. If x satisfies the equation x2-x-1=0, and x<0 then it satisfies
Since 1+x>0, we can do
So by induction:
Hence
If we consider the sequence
its limit is τ.
Similar computations it can be done for the real roots of the polynomial x2-ax-b, with a and b integers:
For example: we have
The positive root of x2-x-n, is
the previous argument shows
In the case of the polynomial x2+x-n, the expression for its positive root is
If we consider x2-ax-b, with a and b positive integer then its positive root is
The previous argument shows the following identity holds:
References:
https://en.wikipedia.org/wiki/Nested_radical
http://mathworld.wolfram.com/NestedRadical.html