The well-known Fibonacci sequence formed by summing consecutive terms looks like:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89……….
And a well known property of this sequence is that if you take the ratio of consecutive terms you get closer and closer to the Golden Ratio the higher up the sequence you go. So if we divide 89 by 55 we get 1.61818181…….
We can create a Fibonacci series incorporating the variable x in the following way:
1, x, (1+x), (1+2x), (2+3x), (3+5x), (5+8x), (8+13x)…
Now a remarkable property of this sequence is that once again, if we take the ratio of consective terms in the sequence, we get the Golden Ratio – Phi – provided x=Phi, which I think is a remarkable result.
What this means is that looking at the sequence with Phi replacing x we get:
1, Phi, (1 + Phi), (1 + 2*Phi), (2 + 3*Phi), (3 + 5*Phi)…
And if taking the ratio of consective terms gives us Phi, this means that:
(1 + Phi) = Phi^2
(1 + 2*Phi) = Phi^3
(2 + 3*Phi) = Phi^4
The first result comes directly from the basic definition of Phi itself, namely that (1 + Phi)/Phi = Phi
Taking a deeper look at the above sequence I was able to come up with a general result, namely:
Phi^n + 2*Phi^(n+1) = Phi^(n+3)