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## 'Golden Fibs' printed from http://nrich.maths.org/

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
$(1+\sqrt 5)/2$.