x = ~ 1.618... or x=~ -.618...
When solving this equation we find that the roots arex = ~ 1.618... or x=~ -.618...
We consider the first root to be Phi. We can also express Phi by the following two series.
Phi = or Phi =
We can use a spreadsheet to see that these two series do approximate the value of Phi.
Other ratios were of interest as well and the third ratio to be considered was the ratio of every third term, or F(n + 3 )/ F(n). The values for the computed ratios for n = 1 to 40 are found in column D. As n increased, it could be seen that there was again a limiting value of approximately 4.23607. This number also has significance with regard to the golden ratio. If we consider the system of equations
x^3 = x^2 + xx^2 = x + 1
and make a substitution, we find that x^3 = 2x + 1. This means that if ß was a solution to the equation x^2 = x + 1, then we should be able to conclude that ß^3 = 2ß + 1. So going through the arithmetic, we find that 2ß + 1= 2.61803. So the next connection that we make is that the ratio ofF(n + 2) / F(n) has a limiting value of ß^3.Finally the fourth ratio to be considered was the ratio of every fourth term, or F(n + 4) / F(n). The values for the computed ratios for n = 1 to 40 are found in column E of the spreadsheet. As n increased, it could be seen that there was a limiting value of approximately 6.8541, and again this number has connections with the golden ratio. If we consider the system of equations