Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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An equation to prove with terms of Fibonacci sequence

I would like to prove an equation but I have stuck. The equation that is to prove is the below: $f(n)^2 + (-1)^{n+1} = f(n+1)f(n-1) , n \ge 2$. I'm trying to do an inductive proof of this equation. ...
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39 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
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6answers
251 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
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4answers
276 views

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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Conjecture: only one even Fibonacci term divided by two gives a prime: $F(9) = 34 = 2 \times 17$

Every Fibonacci term $F(3n)$ is divisible by two $F(3) = 2$ $F(6) = 8$ $F(9) = 34$ $...$ After seeking Fibonacci tables factorization until $F(10000)$, for every term $\frac{F(3n)}{2}$, it ...
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2answers
28 views

How does one arrive at a certain expression for the Fibonacci Zeta function?

In this paper by L. Navas, it is described how one can obtain a analytic continuation of the Fibonacci Dirichlet series (though I'm not sure it's actually a Dirichlet Series). First, the following ...
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79 views

Proving that a Fibonacci number is divisible by integer a

I am working on a review problem and can't figure out how to go about getting to an answer. We are told to let $F_n$ be the $nth$ Fibonacci number (defined as $F_1=F_2=1,F_{n+1}=F_n+F_{n-1}$). Show ...
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2answers
29 views

if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that fibonacci of $d$ divides fibonacci of $n$. i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got ...
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1answer
62 views

Limit of A Fibonacci

Let $P(x)$ be an $n^{th}$ degree-polynomial which is defined below for some odd natural number $n$. And let us denote the set of roots of $P$ by $\{r_1,r_2,r_3,\dots, r_n \}$. $$\displaystyle P(x) = ...
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1answer
68 views

Covering a rectangle of size $n\times1$ with dominos

A rectangle of size $n\times1$ is given. (a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$? (b) In how many ways the rectangle can be covered with ...
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75 views

Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
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4answers
74 views

Why the $GCD$ of any two consecutive fibonnaci numbers is $1$?

Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof. I did the following: $$F_n=F_{n-1}+F_{n-2} \\ ...
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0answers
63 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
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2answers
55 views

Prove that the sum of the first $n$ even terms of the Fibonacci sequence is given by $(F_{3n} + F_{3n + 3} - 2) / 4$ not using induction?

each even term in the Fibonacci sequence has a position index which is a multiple of 3, therefore the even term is $F_{3n}$: $F_{3 (0)} = 0$ $F_{3 (1)} = 2$ $F_{3 (2)} = 8$ $F_{3 (3)} = 34$ $F_{3 ...
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1answer
61 views

Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015

This is a problem that has haunted me for more than a month. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind: Assume that the sequence ...
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2answers
52 views

Proving that $F_{kn}$ is a multiple of $F_n$ by induction on $n$ (Fibonacci numbers)

Question: I want to prove that $F_{kn}$ is a multiple of $F_n$. Approach: I have to deduce this result from the following results: $$F_{n+k} = F_{k}F_{n+1} + F_{k-1}F_{n}$$ I have shown the ...
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2answers
112 views

Sum of odd Fibonacci Numbers

Trying to prove that the sum of odd-index consecutive Fibonacci numbers is the next even-index Fibonacci number. I have a gap in my proof that I cannot figure out. I know that induction would be ...
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1answer
33 views

Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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53 views

Discrete Mathematics Fibonacci Sequence

I am studying for the final exam in my Discrete Mathematics class and came upon the following problem on the study guide we were given. Given the following algorithm: If $n = 0$, then $f(n) = 0$ ...
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1answer
34 views

Why do I get the Fibonacci sequence when I start with 1 and keep using the + sign?

I played with this on a calculator and when I entered 1 and kept hitting the + button, I got the noticeable Fibonacci sequence! Can someone explain to me why this happens?
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2answers
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Proving that for Fibonacci numbers $a_n \lt (\frac {1 + \sqrt 5} 2)^n$ for $n \ge 1$

I'd like to prove that for Fibonacci numbers $a_n \lt \left(\frac {1 + \sqrt 5} 2\right)^n$ for $n \ge 1$. I suppose it needs induction so, after verifying the trivial case $n=1$, the inductive step ...
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4answers
52 views

Finding growth bounds on Fibonacci Sequence

I've been working on this following problem: Find a constant $c< 1$ such that $F_n \leq 2^{cn}$ for all $n \geq 0$. I honestly have no idea where to begin on this. I've done plenty of proofs ...
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0answers
51 views

How to prove a claim about Fibonacci sequence

I have to prove that for any natural number $n$ there exists $i>0$ such that $n\mid F_i$, where $F_i$ is the $i$-th Fibonacci number.
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Is fibonacci sequence a member of more broad family of sequences?

Yesterday, I was pondering on the Fibonacci sequence and I started to discover some features of it that were previously unknown to me. Such as, 1, 1, 2, 3, 5, 8, 13, 21, 34 .... 1 ) The nth element ...
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1answer
122 views

The totient of Fibonacci numbers is divisible by $4$

Let $\{f_i\}_{i\in\mathbb N}$ be the sequence of Fibonacci numbers, i.e. $1,2,3,5,8,13,21,34,55,\cdots$, For every integer $n\gt3$ prove that $$4\mid\phi(f_n)$$ where $\phi$ is Euler's totient ...
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1answer
55 views

Showing that this sum is equal to the fibonacci numbers

How do I show that the following sum is equal to the fibonacci numbers? Atleast numerical evaluation suggests it is $$ \sum_{k=0}^{\lceil n/2\rceil}\binom{n+1-k}{n+1-2k} $$ The image below shows how ...
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1answer
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Generalized Fibonacci Sequences with Modular Arithmetic

Consider the following generalized fibonacci sequence: For $m,p$ positive integers and $g_k =g_k (mod m)$, then for $n=1,2,3,...$ $g_{n+p}=g_{n+(p-1)}+g_{n+(p-2)}+...+g_{n+1}+g_n (modm)$ I need to ...
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1answer
115 views

How do I choose first terms of a Fibonacci sequence?

Let $f(0)=a$ and $f(1)=b$ be the first two terms of a Fibonacci sequence. We know that this sequence is periodic in $\mod{p}$, where $p$ is a prime number, and the period of the sequence is $p-1$. I ...
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A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
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Sum of Power of Two Fibonacci reciprocals

Evaluate $$\sum_{k=0}^{\infty} \frac{1}{F_{2^n}} \;.$$ I'm thinking of using a relation from a term to another.
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Sum of Fibonacci numbers

While trying to find find a formula to calculate the length of the golden spiral I came across the sum of the Fibonacci numbers. I noticed that $$\text{Fibonacci numbers: }1,1,2,3,5,8,13,21,34...$$ ...
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Problem about fibonacci sequence via quadratic roots in gelfand's algebra text.Need hints.

I have solved a preceding question proving that the common ratio of such a sequence is $ \frac {1+\sqrt{5}}{2} $ or $ \frac {1-\sqrt{5}}{2} $ (resolving a quadratic equation) . The present problem is ...
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1answer
80 views

Prove by induction that numbers of Fibonacci of the form $F_{3n}$ are even

I am not quite asking to solve this problem, but I am asking what they are asking me. This is the problem: The Fibonacci numbers $F_n$ for $n \in \mathbb{N}$ are defined by $F_0 = 0$, $F_1 = 1$, ...
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66 views

Fibonacci induction proof?

The Fibonacci Numbers $(f_n)$ are defined $f_1=f_2=1$, and $f_n=f_{n-1}+f_{n-2} ,\,\,\,\forall n \geq2$. Prove that for every integer $n \geq 1$, $$f_1 +f_2 +···+f_n =f_{n+2}−1$$
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Prove the following recurrence: $F_{2n+1}=3F_{2n-1}-F_{2n-3}$

Prove the following identity: $$F_{2n+1}=3F_{2n-1}-F_{2n-3}$$ So far I know that $F_n=F_{n-1}-F_{n-2}\implies F_{2n+1}=F_{2n}+F_{2n-1}$ Just not sure where to go from here to get to the conclusion. ...
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1answer
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Hadamard's product of Fibonacci generating functions.

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.
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1answer
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Fibonacci Number Formula for nth term [duplicate]

Hey is there any known combinatorial formula for nth fibonacci number? (n+1)th fibonacci number is given by summation of r=0 to (round)n/2:C(n-r,r) Can someone verify the formula?Help!
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1answer
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Help with how to prepare the inductive step of a strong induction exercise.

I have the following exercise: "Use strong induction to prove that $f_1^2 + f_2^2 + \cdots + f_n^2 = (f_n)(f_{n+1})$ where $f_n$ in the nth Fibonacci number." This is what I have done: Fibonacci ...
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Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...
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Proving $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $ for Fibonacci numbers

How can I prove the following identity about the Fibonacci numbers by using matrices or determinants? $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $
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Prove that for any power function $f_n = c^{n}$, the number of decimal digits of $ f_{10^n}$ is given by $10^{n}log_{10}c$

I am reading this page about some interesting properties of the Fibonacci numbers: http://mathworld.wolfram.com/FibonacciNumber.html The following is said: The numbers of Fibonacci numbers less ...
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63 views

Given that fib(n)=fib(n-1)+fib(n-2) for n>1 and given that fib(0)=a, fib(1)=b (some a, b >0)

Given that fib$(n)$=fib$(n-1)$+fib$(n-2)$ for $n>1$ and given that fib$(0)=a,$ fib$(1)=b$ $($some $a, b >0)$ which of the following is true? fib$(n)$ is : Select one or more: a. $O(n)$ b. ...
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50 views

Identify fibonacci sequences from a set of data

Let there be a set of increasing order integer data ${a_1, a_2, a_3, a_4, ...}$. given the increasing infinite sequence of integers, how can we determine whether there is an infinite subsequence which ...
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39 views

Fibonacci polynomial summation

The $n^{th}$ value of a polynomial ($S_n$) of order $k$ is a polynomial in $x$ is given by : $\left(\frac{S_n}{x^n}\right)$= $\sum_{j=0}^n \left(\frac{{F_j}^k}{x^j}\right)$ Where ${F_j}$ is the ...
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1answer
21 views

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos where the order matter. Presumably, mathematical induction can be leveraged here. Step 1: Show ...
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2answers
47 views

Applying the mean value theorem to the closed form of the Fibonacci sequence?

Is it possible to apply the mean value theorem to the closed form of the Fibonacci sequence for the 7 numbers starting at 1 and ending with 13 (inclusive)? It's been a LONG time since I studied ...
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1answer
49 views

Deduce a series formula for product of Fibonacci numbers.

Start with the arbitrary pair of Fibonacci numbers $F_{n+1}$, $F_n$ and apply the Euclidean Algorithm to it. Deduce a series formula for the product $F_{n+1}F_n$. I use the formula, $F_{n+1} = F_n + ...
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1answer
59 views

Proving a function by induction [duplicate]

Let $f(n)$ be the function defined by $$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$ How do you prove that $f(n) = f(n+2) - f(n+1)$ ...
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101 views

Relationship between Fibonacci's secuence and $x^2 - x - 1$.

On the end of Apostol's Mathematical Analysis' first chapter, one can find the following exercise (and I paraphrase): Prove that the $n$-th term of the Fibonacci sequence is given by $$x_n = ...
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2answers
155 views

Fibonacci even numbers formula

i found a general formula in any given set of Fibonacci numbers ,to find the next given even number we can use the formula E*4 + Eo where E is the given even number Eo is the even number ...