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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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 ...
7
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4answers
3k views

Show that $f(2n)= f(n+1)^2 - f(n-1)^2$

Let $f(n)=f(n-1)+f(n-2)$ be the Fibonacci sequence with $f(0)=0,f(1)=1$. Show that $$f(2n)= f(n+1)^2 - f(n-1)^2.$$ I have tried several different approaches to this problem. Both inducting from the ...
4
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5answers
806 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
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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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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 ...
16
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4answers
2k views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
4
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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 ...
2
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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 ...
3
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1answer
123 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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2answers
108 views

How find this value $m^2-mn-n^2$

let $$1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots\dfrac{1}{1}}}}}=\dfrac{m}{n}$$ where $m,n$ are positive integer numbers,and such $gcd(m,n)=1$.,and article $1998$ the fractional ...
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3answers
88 views

Trying to prove $\sum_{i=1}^{N-2} F_i = F_N -2$

I'm trying to prove that $\sum_{i=1}^{N-2} F_i = F_N -2$. I was able to show the base case for when $N=3$ that it was true. Then for the inductive step I did: Assume $\sum_{i=1}^{N-2} F_i = F_N -2$ ...
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]{...
8
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2answers
344 views

Fiboncacci theorem: Proof by induction that $F_{n} \cdot F_{n+1} - F_{n-2}\cdot F_{n-1}=F_{2n-1}$

I have the following theorem to prove by induction: $$ F_{n} \cdot F_{n+1} - F_{n-2}\cdot F_{n-1}=F_{2n-1} $$ It is mentioned in my script that the proof should be possible only by using the ...
2
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2answers
560 views

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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3answers
145 views

Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$

The Fibonacci sequence $F_0, F_1, F_2,\dots$ is defined by the rule $F_0=0, \ F_1=1, \ F_n = F_{n−1} + F_{n−2}$. Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$. So for the base case: ...
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3answers
182 views

Prove $F(n) < 2^n$ [closed]

Consider the Fibonacci function $\large{F(n)}$, which is defined such that $F(1) = 1, F(2) = 1$, and $F(n) = F(n−2)+F(n−1)$ for $n > 2$ I know that I should do it using mathematical induction but ...
4
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1answer
78 views

Fibonacci Numbers, show $F_n \ge 2^{n/2}$ for $n \ge 6$. [duplicate]

I want to show that for the Fibonacci numbers, $F_n$ $>=$ $2^{n/2}$ for n $>=$ 6. My thought was to prove this via induction. I showed the base case is true for $F_n$, n=6 and 7. I assumed ...
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2answers
95 views

Proof by mathematical induction - Fibonacci numbers and matrices

Using mathematical induction I am to prove: $ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)^n $ = $ \left( \begin{array}{ccc} F_{n+1} & F_n \\ F_n & F_{n-1} \end{...
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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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2answers
416 views

The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer

For each $n$ consider the expression $$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$$ I am trying to prove by induction that this is an ...
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3answers
230 views

Fibonacci Sequence or Golden Ratio?

Using the polar coordinate system, $r$ increases directly with $\theta$. In other words, $r=k\theta$. Which of the following shapes is constructed? A) Fibonacci Sequence B) Golden Ratio C) ...
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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 ...
2
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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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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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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 ...
1
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1answer
49 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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0answers
44 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
6
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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,\...
21
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17answers
27k views

Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
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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 ...
12
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3answers
2k views

How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$

How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$ where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?
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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) =...
9
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3answers
1k views

Proof of this result related to Fibonacci numbers: $\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$?

$$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$$ Somebody has any idea how to go about proving this result? I know a proof by ...
3
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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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3answers
155 views

Finding $a \bmod b$ where $a,b$ are large Fibonacci numbers

For moderately large values of $b$ we can use Chinese Remainder Theorem, by factorizing $b$. But for very large values of $b$, (for example $b$ is the 1000th Fibonacci number) factorization will take ...
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2answers
176 views

Remainders of Fibonacci numbers

Let $a>b$ be positive integers. Is there a Fibonacci number that is $b$ modulo $a$? We know that the Fibonacci numbers are periodic modulo $a$. Indeed, consider pairs $(F_i,F_{i+1})$ modulo $a$. ...
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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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2answers
150 views

Inductive proof of the identity $\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$ [duplicate]

I'm trying to prove the following identity: $$\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$$ I need to prove it using induction (not a counting argument), I ...
3
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4answers
318 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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0answers
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 ...
7
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3answers
161 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 ...
0
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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 ...
0
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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 ...
0
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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 ...
1
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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 ...
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 $β=\dfrac{1-\sqrt{5}}{2}.$ ...
0
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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} = ...