What is the relationship between the fibonacci series and the golden ratio?

4 Answer(s)

limit n->∞ f(n+1) / f(n) = φ

There are many including:

ϕⁿ = ϕF(n) + F(n-1)
e.g., ϕ⁰ = ϕ*0 + 1 = 1, ϕ¹ = ϕ*1 + 0 = ϕ, ϕ² = ϕ*1 + 1 = ϕ+1, etc.

and

F(n) = (ϕⁿ - (-ϕ)⁻ⁿ))/√5
e.g., F(1) = (ϕ¹ - (-ϕ)⁻¹))/√5 = (ϕ - (1-ϕ))/√5 = (2ϕ - 1)/√5 = √5 / √5 = 1
e.g., F(2) = (ϕ² - (-ϕ)⁻²))/√5 = (ϕ+1 - (2-ϕ))/√5 = (2ϕ - 1)/√5 = √5 / √5 = 1
e.g., F(3) = (ϕ³ - (-ϕ)⁻³))/√5 = (2ϕ+1 - (3-2ϕ))/√5 = (4ϕ - 2)/√5 = 2√5 / √5 = 2, etc.

idk