Ratio of Fibonacci numbers

Question

I have observed the following property: If $F_n$ denotes the $n^{th}$ Fibonacci number and $F_1=1; F_2=1$ Then $\frac{F_{2n+1}}{F_{2n}}>\phi$ And $\frac{F_{2n}}{F_{2n-1}}<\phi$ For all natural numbers n. How do I prove this? Here $\phi$ is the golden ratio

Answer 1

Hint:You can use the general form of the fibonnaci sequence

$$ F_n={({1+\sqrt{5}\over2})^n-({1-\sqrt{5}\over2})^n\over\sqrt{5}}={\phi^n-(-{1\over\phi})^n\over\sqrt{5}} $$