Linked Questions
19 questions linked to/from How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
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Proof the golden ratio with the limit of Fibonacci sequence [duplicate]
Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$
The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$.
Sorry as can be proceed??
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Limit of the ratio of consecutive Fibonacci numbers [duplicate]
I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
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Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]
Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists.
How can we prove ...
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3
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Show that the limit exists and find it's value? [duplicate]
The Fibonacci series defined recursively by $x(1) = 1, x(2) = 2$ and
$x(n+1) = x(n) + x(n-1)$
Find$$\lim_{n\rightarrow\infty}\frac{x(n+1)}{x(n)}$$
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1
vote
5
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Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]
We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider
$$
\lim_{n \to \infty} \frac{a_{n+1}}{a_n}.
$$
I wrote a script and found that this ...
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5
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2
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Proving that ratio of two consecutive Fibonacci numbers to converges to golden ratio by induction [duplicate]
$$\varphi = \frac{1 + \sqrt{5}}{2}$$
We want to prove that ratio of two consecutive Fibonacci numbers approaches $\varphi$ by induction and also utilizing Newton-Raphson method for approximating $\...
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7
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How to prove that the Binet formula gives the terms of the Fibonacci Sequence?
This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$.
Show that
...
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10
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Mathematical function that converges towards $7$?
My friends and I are finishing High School in Denmark. We have to do a math poster for some school activity, where the poster needs to have something to do with the number $7$. So my question is: does ...
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4
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3
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Solution to $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$ [duplicate]
Possible Duplicate:
Why does this process, when iterated, tend towards a certain number? (the golden ratio?)
Please post your favorit solution to the following
Compute $x=1+\cfrac{1}{1+\cfrac{1}...
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1
answer
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Another way to go about proving the limit of Fibonacci's sequence quotient.
It is not difficult to inductively prove that
$$\eqalign{
& \phi = \phi + 0 \cr
& {\phi ^2} = \phi + 1 \cr
& {\phi ^3} = 2\phi + 1 \cr
& {\phi ^4} = 3\phi + 2 \...
- 122k
2
votes
3
answers
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Binet's formula to golden ratio
Given that the Binet's formula is $$F_n = \frac{(1+\sqrt 5)^n - (1-\sqrt 5)^n}{2^n\sqrt 5}$$ I want to verify that $$ \lim_{n\to \infty}\frac{F_{n+1}}{F_n} = \frac{1+\sqrt 5}{2} $$
I reached this ...
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2
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3
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If $x_0=1$ and $x_n=\frac {1}{1+x_{(n-1)}}$, find: $\lim_{x\to\infty} x_n$ [duplicate]
If $x_0=1$ and $x_n=\dfrac {1}{1+x_{(n-1)}}$, find $\displaystyle\lim_{x\to\infty} x_n$.
My attempt:
$x_1=1+\dfrac 1 2=\dfrac 3 2$
$x_2=1+\dfrac {1}{1+\frac 3 2}=\dfrac2 5$
Which gives following ...
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0
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3
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Fibonacci sequence/recurrence relation (limits)
Let $\lbrace F_n\rbrace_{n \in \mathbb{N_0}}$ be the Fibonacci sequence.
$F_{n+1}=F_{n-1}+F_{n-2}$ for $n \in \mathbb{N}$ with $n \geq 2$ and start values $F_0:=0$ and $F_1:=1$.
How to determine:
$...
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2
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3
answers
190
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Ratio of odd terms in fibonacci sequence
I think that the sequence $$R_n=\frac{f_{2n+1}}{f_{2n-1}},$$ (where $f_n$ is the fibonacci sequence) converges to $\varphi^2$ where $\varphi$ is the "golden ratio." A quick check with the calculator ...
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Two ٍEquations Defining the Golden Ratio Differently
In most sources, the Golden Ratio has been referred to as: $\phi = \frac{\sqrt{5}+1}{2}$. Nevertheless, in the book " The Theory of Numbers, a Text and Source Book of Problems ", by Prof. Andrew Adler ...
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