Golden Fibs

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative


The Fibonacci sequence $F_n$ is defined by the relation $$F_{n+2}=F_n + F_{n+1}$$ where $F_0=0$ and $F_1=1$. Now suppose that we take the same relation and more general sequences $X_n$ with any two starting values $X_0$ and $X_1$. Prove that the sequence is geometric if and only if the first two terms are in the ratio $1 : \pm \phi$ where $\phi$ is the golden ratio $(1+\sqrt 5)/2$.