### 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.