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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Closed form expression for zero of recurrence relation

Given the recurrence $d(i+1)=xFib(2i+1)-nFib(2i)$, where $Fib$ denotes the Fibonacci sequence (i.e. $Fib(0)=0, Fib(1)=1, Fib(2)=1, Fib(3)=2$, etc) and $n$ and $x$ are arbitrary integers, is it ...
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2answers
46 views

How to prove this sequence is null?

I am working on the fibonacci numbers series using the ratio. To prove convergence I want to show that the sequence of the series is going to 0. And then according to the Leibniz criterion the series ...
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0answers
37 views

Fibonacci numbers and binomial theroem [duplicate]

So I am trying to prove $$\sum_{i=0}^n{nCi×F_i} = F_{2n}$$ Such that $$nCi = \frac {n!}{i!×(n-i)!}$$ And $F_i$ is the ith value of the fibonacci sequence such that $F_0 = 1$ and $F_1 = 1$ I have ...
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1answer
60 views

Proof: Fibonacci Sequence (2 parts)

Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers. I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely ...
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0answers
31 views

Divisibility of Fibonacci Sequence mod prime

I have to solve the following problem and I have a few questions: Consider the Fibonacci sequence defined as $F_n:=2F_{n-1}+F_{n-2}$ with $F_0=1$ and $F_1=1$. Now, I need to prove that for any odd ...
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3answers
119 views

Cesaro identity for Fibonacci numbers

I am stuck with the identity $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$ which happens to be formula 80. I am using induction, but so far without too much result. $$ \sum_{k=1}^{n+1} \binom{n+1}{...
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1answer
342 views

Given Two Fibonacci numbers, predicting the median Fibonacci number

Wolfram Alpha gives the $100$th fibonacci number to be $354224848179261915075$ and the $104$th fibonacci number to be $2427893228399975082453$. Just from this, can we deduce what the $102$th fibonacci ...
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1answer
56 views

How many times can $p$ divide $F_n$?

Given a prime $p$ and a number $n$ (or perhaps just an upper bound $x$ with some unknown $n\le x$), trivially one has $$ \operatorname{ord}_p F_n\le\frac{\log F_n}{\log p}\approx\frac{n\log\varphi}{\...
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2answers
37 views

Sum of Squares for Odd Fibonacci Numbers

I am trying to prove the following theorem by induction: THEOREM: For the Fibonacci sequence $F_1$, $F_2$, ... , $F_n$ defined as, $F_1$ = $F_2$ = 1 $F_n$ = $F_{n-1}$ + $F_{n-2}$ for n >= 3, For ...
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1answer
125 views

Proving that every integer has a Fibonacci number multiple

Show that for any positive integer, there exists a Fibonacci number N such that N is divisible by the integer. I'm not really sure how to begin my approach to this problem, would really appreciate ...
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1answer
31 views

Prove that $F_n < 2^n$ for every $n \geq 0$ - Mathematical induction

The Fibonacci sequence $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$, ... is defined as a sequence whose two first terms are $F_0=0$, $F_1=1$ and each subsequent term is the sum of the two previous ones: $...
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0answers
37 views

$|x^2-xy-y^2|=1$ implies that $x=\pm F_{n+1},\; y=\pm F_n$

So I've proved that $ A= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \implies A^n= \begin{pmatrix} F_{n+1} & F_n\\ F_n & F_{n-1} \end{pmatrix} $ for Fibonacci numbers $F_i$. I'm ...
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1answer
43 views

How to apply geometric series concepts into these numbers?

This is a basic level question and it is homework for someone that I am trying to help out. It is indicated as a Fibonacci puzzle. But I am not able to fit the numbers into a general geometric formula....
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2answers
66 views

Proving a Fibonacci identity: $F_{2n} = F_n (F_{n+1} + F_{n-1})$

$$ F_{2n} = F_n (F_{n+1} + F_{n-1}) $$ I'm so stuck. I've used the definition of Fibonacci to change $F_{2n+2}$ into $F_{2n+1} + F_{2n}$. Can't use other properties, only the inductive hypothesis and ...
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1answer
50 views

Clarification on tribonacci numbers exercise

From what I know the Tribonacci sequence is given by: T(n) = T(n-1) + T(n-2) + T(n-3) My book says that "We can show by induction that for large enough n, the Fibonacci numbers satisfy the ...
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1answer
54 views

Are any factors of Lucas numbers divisible by a Fibonacci number greater than three?

The congruence relation for Fibonacci and Lucas numbers is stated: If Fn > 3 is a Fibonacci number then no Lucas number is divisible by Fn. However, does this apply to the factors as well?
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0answers
45 views

Fibonacci numbers notation

$F_{i}$ denotes the $i^{th}$ Fibonacci number, but what does it mean when there are 2 subscripts, $F_{ij}$? Context: show that $F_{i}$|$F_{ij}$ (where $i$ and $j$ are positive integers) Thanks
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1answer
86 views

Summation of a series involving powers of Fibonacci numbers.

I'm interested in this series: $$\mathcal S_p=\sum_{n=1}^\infty\frac{\left(F_n\right)^p}{2^{np}},\quad p\in\mathbb N,\tag1$$ where $F_n$ are the Fibonacci numbers, defined by the recurrence $$F_1=1,\...
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1answer
31 views

Fibonacci clarification

As explained in Excursion 4.5, the Fibonacci numbers are defined by the rules: F(0) = 0, F(1) = 1, and for all n with n ≥ 2, F(n) = F(n-1) + F(n-2). Which of these claims about the Fibonacci numbers ...
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2answers
61 views

Induction proof fibonacci numbers

I need to prove the following with induction: n∑i=1 F(2i-1) = F(2n) for all n >= 1 I am stuck in my inductive step: n∑i=1 F(2i-1) = n∑i=1 F(2i-1) + F(2(n + 1) -1) = F(2n) + F(2(n + 1)-1) =...
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2answers
234 views

Fibonacci and Matrices [duplicate]

Consider Matrix $$ A = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix} $$ Investigate the sequence of powers of $A$ (i.e. $A^n$ for $n = 1, 2, 3, 4,\ldots$. Verify that $$A^n = \begin{pmatrix}F_{...
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2answers
53 views

Convergence of Binet's formula expression for Fibonacci

Let $\displaystyle \phi = \frac{1+\sqrt{5}}{2}$ and $\displaystyle \psi = \frac{1-\sqrt{5}}{2}$. Consider the Fibonacci sequence defined by: $$ \displaystyle a_n = \frac{\phi^n - \psi^n}{\sqrt{5}} $$ ...
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4answers
321 views

Fibonacci sequence divisible by 3?

I have a recursion question for my combinatorial class. I'm looking at the Fibonacci sequence $f(n)=f(n-1)+f(n-2)$ for $n \geq 3$ with $f(1)=f(2)=1$. I'm trying to prove that $f(n)$ is divisible by 3 ...
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2answers
84 views

Source and/or combinatorial interpretation for $F_{n+k} = \sum_{i=0}^{k} \binom{k}{i}F_{n-i}$

Through some fussing with Taylor's Theorem in the discrete calculus described here (among other places), I found what I believe to be an identity: $$F_{n+k} = \sum_{i=0}^{k} \binom{k}{i}F_{n-i}$$ ...
7
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1answer
85 views

Is there a non-constant function $f$ such that $f'(x) = f(x - 1)$?

In discrete calculus, where the difference operator $\Delta f = f(x + 1) - f(x)$ takes the place of $\frac{d}{dx}$, Fibonacci sequences are given by the functions satisfying: $$ \Delta f(x) = f(x - 1)...
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1answer
57 views

Sum of $i$ times $(i-1)^\text{th}$ Fibonacci Number [closed]

Consider the expression $$\sum\limits_{i=1}^n i \cdot F_{i-1}$$, where $F_{0}=0, F_{1}=1, F_{2}=1, F{3}=2,$ etc. Is there a closed formula for this? If so, find it.
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42 views

Finding the n-th Pisano Period (for small n)

From Wikipedia: [...] the $n$th Pisano period, written $\pi(n)$, is the period with which the sequence of Fibonacci numbers taken modulo $n$ repeats. For example, the Fibonacci numbers modulo 3 ...
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2answers
35 views

Proving an identity of Fibonacci Numbers by induction

Say we know this as a given: $$E_0 = A$$ $$E_1 = B$$ $$E_2 = A + B$$ $$E_3 = A + 2B$$ $$E_4 = 2A + 3B$$ $$E_5 = 3A + 5B$$ $E_{n+1}$ is defined as: $$E_{n+1} = E_n + E_{n-1}$$ You can start to see ...
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1answer
72 views

Prove that if Euclid's algorithm applied to the pair $(u,v)$ takes $n$ steps, then $u \geq f_{n+2}$ and $v \geq f_{n+1}$

Let $u, v \in Z^+$ satisfy $u > v$. Prove that if Euclid's algorithm applied to the pair $(u,v)$ takes $n$ steps, then $u \geq f_{n+2}$ and $v \geq f_{n+1}$. Where the $f_n$ values refer to the ...
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1answer
49 views

Prove that for all $n \in Z^+$, Euclid’s algorithm applied to the pair $(f_{n+1}, f_{n+2})$, takes $n$ steps.

Prove that for all $n \in Z^+$, Euclid’s algorithm applied to the pair $(f_{n+1}, f_{n+2})$, takes $n$ steps. (Here, as previously, $f_n$ denotes the nth Fibonacci number.) I don't understand how to ...
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0answers
35 views

Periods of Fibonacci numbers in mod. Number Theory

Work out the periods $π(n)$ of the $\mod n$ such that $$f_k ≡ f_{k+π(n)} \mod n$$ I got $π(2)=3$,$π(3)=8$,$π(4)=6$ by computing it and looking at the periods. Now Part 2 Prove that for all ...
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3answers
163 views

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$.

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. Assume $\binom{n}{k} = 0$ if $k>n$. Does anyone know an elementary ...
2
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2answers
51 views

Using induction to prove a formula for the Fibonacci sequence involving the solutions of $x^2=x+1$

Let $\{f(n)\}_{n=1}^{\infty}$ denote the Fibonacci sequence defined by $f(1)=1, f(2)=1$, and $f(n)=f(n-1)+ f(n-2)$ for all $n\geq 3$. Let $α=\dfrac{1+\sqrt{5}}{2}$ and $β=\dfrac{1-\sqrt{5}}{2}.$ ...
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1answer
42 views

Proving Fibonacci sequence with mathematical induction

Okay, so I have the following thing: $$\sum_{i=1}^a F_{2i}=F_{2a+1}-1 $$ It's to do with Fibonacci sequence. I can do the basis step of MI fine (proving for $a = 1$) However the inductive step has ...
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1answer
72 views

Mathematical induction proof of $\sum_{i = 1}^{n} F_{2i} = F_{2n + 1} - 1$

Use Mathematical Induction to show that $$\sum\limits_{i=1}^n F_{2i}=F_{2n+1}-1$$ for all integer $n\geq1$. My answer: Base case: for n = 1 $$\sum_{i = 1}^{n} F_{2i} = \sum_{i = 1}^{1} F_{2i} = ...
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3answers
87 views

Proof on Fibonacci sequence: $F(1) + F(3) + \cdots + F(2n-1) = F(2n)$ using induction and recursion

The problem is: Use induction and the recursive formula to prove that: $$F(1) + F(3) + \cdots + F(2n-1) = F(2n)$$ For the base case I let $n=1$ which gave $$F(1) = F(2(1))$$ $$1=1$$ Which is ...
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1answer
90 views

How to find pythagoras triplet using the fibonacci sequence?

I'm using the Fibonacci sequence to generate some Pythagorean triples $(3, 4, 5,$ etc$)$ based off this page:Formulas for generating Pythagorean triples starting at "Generalized Fibonacci Sequence". ...
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1answer
33 views

Fibonacci Sequence Squared

I have been learning about the Fibonacci Numbers and I have been given the task to research on it. I have been assigned to decribe the relationship between the photo (attached below). I know that the ...
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1answer
77 views

Consider a recursive sequence. Find all values of x for which this sequence is bounded

Consider a recursive sequence $a_{n+1} = a_n + a_{n-1}$ for all $n \geq 2$ with $a_{1} = 1$ and $a_2 = x$. Find all values of $x$ for which this sequence is bounded.
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1answer
65 views

Is there a formula to calculate factors of the smallest integer u, for which n, divides a Fibonacci number?

I have read that a conjecture for Fibonacci entry points, by Paul Bruckman and Peter Anderson has been proven for prime p, that uses the Galois theory and the Chebotarev density theorem to compute the ...
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0answers
62 views

Does the p-adic valuation of n, apply to Fibonacci numbers and Fibonacci-Wieferich primes?

Does the p-adic valuation of n, apply to Fibonacci-Wieferich primes in the following way? Let Fup be the smallest Fibonacci number divisible by a prime p > 5, then Vp(Fupk) = Vp(Fup) + Vp(k). The ...
4
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1answer
64 views

If $\frac{p_{n+1}}{np_n} \to p > 0 $, then $\sqrt[n+1]{p_{n+1}}-\sqrt[n]{p_{n}} \to \frac{p}{e}$

Problem: Prove that, if a sequence ${p_n}$ satisfies $p_n > 0$ and $\lim\limits_{n \to \infty} \frac{p_{n+1}}{np_n} = p > 0 $, then $\lim\limits_{n \to \infty} \left(\sqrt[n+1]{p_{n+1}}-\sqrt[n]{...
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1answer
58 views

The sum of the Reciprocal of the Partial Sum of the Consecutive Fibonacci Numbers Series [closed]

How to prove that this conjecture is true for $n$th order Fibonacci number: $$1\le\sum_{n=1}^\infty\dfrac{1}{F_{n+2}-1}<2.5$$
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2answers
58 views

Calculating number of tile sequences

My daughter (aged 12) came to me with the problem below. I was able to help her to some extent but I could not see an age-appropriate solution. That is, I could imagine solutions involving factorials /...
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1answer
55 views

Are Fibonacci numbers with a square prime index always divisible by $F_p$?

I am doing some research on sequences and I need some help. The sequence of $F_{p^2}$ seems sort of different. It seems that because the index only has one distinct prime factor, as a result the only ...
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1answer
59 views

Showing $P_n = {F_n \over {2^n}}$

Let $P_n$ be the probability that, if you flip a fair coin $n$ times, there are no consecutive heads. Also, let $F_n$ be the $n^{th}$ Fibonacci number, normalized by $F_1 = 1$ and $F_2 = 2$ and $F_n = ...
1
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1answer
78 views

A Property of Fibonacci Numbers [duplicate]

I've seen the property $$f_{n+1} f_{n−1} = f_n^2 + (−1)^n, n ≥ 2.$$ for Fibonacci numbers at Abstract Algebra book of Thomas W. Judson. I've tried it for a few Fibonacci number, and I've really ...
1
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1answer
37 views

Fibonacci sequence developing [duplicate]

For the sum $$\sum_i^n {n-i \choose i}$$ I evaluate it for $n=1,2,3,4,5$ For $n=1$ we have $$\sum_{i=0}^1 {1-i \choose i} = {1 \choose 0} + {0 \choose 1} = 1 + 0 = 1$$ For $n=2$ we have $$\sum_{i=0}^...
3
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1answer
49 views

Showing $F_{\frac{p^2+1}{2}}\equiv p-1 \pmod{p}$ when $p\equiv \pm 2 \pmod{5}$ and $p\equiv 3 \pmod{4}$

A while back I was messing around with representations of finite fields and found this problem above while doing so. I'll explain below how I came to this point but my question is: Question: How ...
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5answers
63 views

Proof about specific sum of Fibonacci numbers

Let $F_k$ denote the $k$-th Fibonacci number. Find a formula for and prove by induction that your formula is correct for all $n > 0$. $$ (-1)^0 F_0+(-1)^1 F_1+(-1)^2 F_2+\cdots+(-1)^n F_n=\ ? $$ I ...