The Fibonacci sequence $F_n$ is defined by the relation

${\displaystyle F_{n+2}=F_n+F_{n+1}}$

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 $1:\pm \varphi$ where $\varphi$ is the golden ratio $(1+\sqrt{5})/2$.