# Recursive sequence (Fibonacci Sequence) & Limits

• Mar 1st 2010, 08:59 AM
darksoulzero
Recursive sequence (Fibonacci Sequence) & Limits
Hey, I would like some help with this question:

A recursive sequence is a sequence where the $\displaystyle n$th term, $\displaystyle t_{n}$, is defined in terms of preceding terms, $\displaystyle t_{n-1}, t_{n-2}$, etc

one of the most famous recursive sequences is the Fibonacci sequence, created by Leonardo Pisano (1170-1250). The terms of this sequence are define as follows $\displaystyle f_{1} = 1, f_{2} = 1, f_{n} = f_{n-1} + f_{n-2}$ where $\displaystyle n \geq 3$

It asks me to graph some things, did those, but then it asks me to write an expression, using a limit, to represent the value of the ratios of consecutive terms of the Fibonacci sequence. How would I go about doing this?

• Mar 1st 2010, 11:13 AM
running-gag
Hi

$\displaystyle f_{n} = f_{n-1} + f_{n-2}$

$\displaystyle \frac{f_{n}}{f_{n-1}} = 1 + \frac{f_{n-2}}{f_{n-1}}$

$\displaystyle \frac{f_{n}}{f_{n-1}} = 1 + \frac{1}{\frac{f_{n-1}}{f_{n-2}}} = 1 + \frac{1}{1 + \frac{1}{\frac{f_{n-2}}{f_{n-3}}}}$
• Mar 1st 2010, 11:57 AM
chisigma
Setting $\displaystyle \rho_{n} = \frac{f_{n}}{f_{n-1}}$ the sequence of the $\displaystyle \rho_{n}$ is generated recursively by the relation...

$\displaystyle \rho_{n}= 1 + \frac{1}{\rho_{n-1}}$ (1)

... that is equivalent to the difference equation...

$\displaystyle \Delta_{n} = \rho_{n} - \rho_{n-1}= 1 - \rho_{n-1} + \frac{1}{\rho_{n-1}}$ (2)

The (2) indicates that the limit when n tends to infinity of the $\displaystyle \rho_{n}$ is given by the positive solution of the equation...

$\displaystyle \rho^{2} - \rho - 1 = 0$ (3)

... so that is...

$\displaystyle \lim_{n \rightarrow \infty} \rho_{n} = \frac{1 + \sqrt {5}}{2} = \varphi = 1.61803398874989\dots$ (4)

The constant $\displaystyle \varphi$ is known as 'golden ratio'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$