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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 $(1+\sqrt 5)/2$.
Golden Fibs
Age 16 to 18
Challenge Level 
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 $(1+\sqrt 5)/2$.
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