"Golden" geometric progression

But we can got even greater "aesthetic enjoying", if we execute the following transformation over the identity:

(1)

Let's multiply all terms of the identity (1) by the "golden" proportion t, and then let's divide them by t. As a result we will get two new identities:

(2)
or
(3)

If we continue to multiply the identity (2) by t, and to divide the identity (3) by t and to strive this process ad infinitum, we will come to the following elegant identity connecting the next degrees of the golden proportion:

(4)

where a number of n is integer and takes its values from the set: 0, ±1, ±2, ±3, ... .

The identity (4) can be expressed by the following words: "Any integer degree of the golden proportion equals to the sum of two previous ones".

This property of the golden proportion is really "unique"! Really, it is very difficult to imagine that the following identity is "absolutely valid":

(5)

But its validity uniquely follows from validity of the identity (4).

Moreover. The following identity is absolutely valid:

and there exists an infinite number of similar identities for the number t100.

Let's consider the series of the golden proportion degrees, that is:

(6)

The sequence (6) has rather interesting mathematical property. On the one hand, the sequence (6) is "geometrical progression" because each term of (6) is equal previous one multiplied by constant for the given progression number of t called as a denominator of geometrical progression, that is:

(7)

On the other hand, according to (4) the sequence (6) is the arithmetic series because each term of it is equal to the sum of two previous terms. Let's note, that the property (4) is characteristic only for geometrical progression with the denominator of t and such geometrical progression is called as the "golden" progression.

In geometry each geometrical progression of the type (6) meets some equiangular spiral. In opinion of many researchers, the property (4), which is inherent only in the "golden" progression is the cause of widespread occurrence just of the "golden" equiangular spiral in the forms and structures of the alive nature. But about it we will talk later, when we will consider applications of the "golden" spirals in the alive nature.