Request #03
Using this formula for the ratio, q:
a / b = (a + b) / a
We have:
q := a / b (* for lhs *) (a + a / q) / a = (* eliminating b by substitution in rhs *) 1 + 1 / q
Therefore:
q = 1 + 1 / q q - 1 - 1 / q = 0 q**2 - q - 1 = 0 q**2 - q + 1/4 = 5/4 (* completing the square: + 5/4 *) q = sqrt(5/4) + 1/2 (* q**2 - q + 1/4 = (q - 1/2) ** 2, sqrt *) 1.618034 ← q = (sqrt(5) + 1) / 2
Incidentally, we have:
q - 1 = 1 / q
Whenever you are thinking about a natural rate of growth, you can approximately take 62 percent, or, adding a bit more of precision, 61.8 percent. Or, if you want to calculate the next step in a very efficient way, you could use 0.625, which is 5/8. Though, there is a much more straightforward way for generating this sequence, coming back to, next.