Consider the Fibonacci Sequence recursively defined by F_{n+2}=F_{n+1}+F_n for all n\geq 1 with F_1=F_2=1 . Define another sequence u_n=\frac{F_{n+1}}{F_n} . Also the Golden ratio \phi is the positive root of \phi^2-\phi-1=0 and \psi is the negative root. Which of the following are True?

Question

Consider the Fibonacci Sequence recursively defined by   F_{n+2}=F_{n+1}+F_n  for all   n\geq 1  with   F_1=F_2=1 . Define another sequence   u_n=\frac{F_{n+1}}{F_n} . Also the Golden ratio    \phi  is the positive root of   \phi^2-\phi-1=0  and   \psi  is the negative root.  Which of the following are True?

Accepted Answer

u_n  are the continued Fraction Approximations of   \phi .

Accepted Answer

F_n=\frac{\phi^n-\psi^n}{\phi-\psi}

Accepted Answer

u_n  converges to   \phi

Answer

u_n  is decreasing and bounded below by   \phi  for odd   n_ .

Answer

u_n  is increasing and bounded above by   \phi  for even   n_ .

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Consider the Fibonacci Sequence recursively defined by F_{n+2}=F_{n+1}+F_n fo...

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