Is this equation the origin of the metallic means or silver means? (also ratios or constants)
$ 2^2=(x+1/x)^2−(x−1/x)^2 $
As discussed in my previous question, we can link this equation with Pythagoras’s theorem, so that for each number (x) we have a corresponding right angled.
If B = 1, then C = √5 (Square root of 5), and because the following is true, we can calculate x:
$ 2(1/x) = C-B $
and,
$ 2(x) = C+B $
Which gives us these two equations :
Which we can combine into one equation :
If we do the same for B = 1-9, we get the following:
Each of these has a corresponding sequence of numbers, as you can see the first and second sequences are called Fibonacci and the Pell sequence. These sequences are known as the metallic means or silver means.
As you can see below, as these sequences approach infinity, the ratio of the last two consecutive numbers tends towards the number (x).
As you can see below, as these sequences approach zero, the ratio of the last two consecutive numbers tends towards the number (1/x).
Why are the sequences above not mentioned on the Wikipedia page metallic means? Including zero, we get these simplified equations:
Is this equation the origin of the metallic means ? And if so, why is it not on the wiki page? Can anybody help or provide more information?
$ 2^2=(x+1/x)^2−(x−1/x)^2 $
It is assumed that (x) is always ≥ 1 and the (1/x) is always ≤ 1.