Golden Number and Lucas Sequence

The golden number is

There is extensive literature on the golden number which is also known as the golden ratio. The exposition in Wolfram Mathworld [http://mathworld.wolfram.com/GoldenRatio.html] is excellent. John Baez gave an interesting exposition [http://math.ucr.edu/home/baez/week203.html]. Mario Livio’s book “The Golden Ratio” is popular.

There is a property of the golden number that does not get enough attention. The general form of this property is

where n=1,2,3,4,5,… are the counting numbers and Ln are the Lucas numbers. Note that n starts from 1.

The Lucas numbers are defined by the recurrence relation

(Fibonacci numbers have the same recurrence relation but it is a different sequence 0,1,1,2,3,5,8,13,….)

Explicitly, the first five representations of this wonderful property of the golden number are:

Note that these are not equations. These are mathematical identities. The term “equation” is used when we have variables. An equation relates different variables to each other. In these identities we have a number (not variable) on the left hand side and equivalent number on the right hand side. The golden number $\Phi$ is not a variable. It is a number.