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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-2
votes
2answers
13 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
7
votes
2answers
63 views

A number $N$ is a $k$-nacci number if and only if …

For $k\ge 2\in\mathbb N$, one can define the $n$-th $k$-nacci number $f_k(n)\ (n=0,1,\cdots)$ as $$f_k(0)=f_k(1)=\cdots=f_{k}(k-2)=0,\ \ ...
35
votes
1answer
1k views

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
2
votes
1answer
15 views

How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
-5
votes
1answer
17 views

is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$ and vice versa? [on hold]

Is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$? Also is there constant $k$ that $k2^n>F_n$?
-3
votes
1answer
28 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences. [on hold]

Let $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that $c\phi^n + ...
1
vote
2answers
34 views

Limit and Fibonacci [on hold]

How to prove $${\lim_{n \to \infty} \frac{F_{kn}}{{F_n}^k} = 5^{(k-1)/2}}$$ Non-induction method is prefered.
1
vote
1answer
24 views

Are there 3D geometric proofs of Fibonacci identities?

There is a significant number of identities involving Fibonacci numbers that can be proven in a geometric way, like this one: However, I couldn't find any that involves 3D geometry. I also couldn't ...
0
votes
0answers
59 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
1
vote
2answers
116 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
1
vote
1answer
36 views

Calculating the hitting probability using the strong markov property

** This problem is from Markov Chains by Norris, exercise 1.5.4.** A random sequence of non-negative integers $(F)n)_{n\ge0}$ is obtained by setting $F_0=0$ and $F_1=1$ and, once $F_0,\ldots,F_n$ are ...
1
vote
0answers
45 views

Proving an equation dealing with Fibonacci numbers

Prove that: $f(2 \cdot k) = f(k) \cdot  f(k + 1) + f(k - 1)  \cdot f(k) $ Where $f(k)$ is the kth Fibonacci number. Also prove that: $f(2 \cdot  k + 1) = f(k) \cdot f(k) + f(k + 1) \cdot f(k + 1) ...
2
votes
4answers
78 views

A calculus proof for the general term of the Fibonacci sequence

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
2
votes
4answers
115 views

Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$ [closed]

Show that $$f_{n+1}f_{n-1}-f_n^2=(-1)^n$$ when $n$ is a positive integer and $f_n$ is the $n$th Fibonacci number.
1
vote
0answers
112 views

using induction to prove that the formula for finding the n-th term of the Fibonacci sequence is: [duplicate]

May someone help me? I am trying to use induction to prove that the formula for finding the $n^{th}$ term of the Fibonacci sequence is: ...
1
vote
2answers
103 views

Inductive proof of a formula for Fibonacci numbers

May someone help me? I am trying to use induction to prove that the formula for finding the $n$-th term of the Fibonacci sequence is: ...
2
votes
3answers
182 views

Statement on the Fibonacci sequence

Question: Let $n,m,\in\mathbb{N^*}$ with $n>1$ and let $u_n$ denote the $n$-th term of the Fibonacci sequence, then $$u_{n+m}=u_{n-1}u_m+u_nu_{m+1}$$ I know these theorems: Two consecutive ...
0
votes
1answer
98 views

Two issues of Number Theory

Knowing in Fibonacci sequence$$u_n\mid u_m\Longleftrightarrow n\mid m$$ Question 1: In Fibonacci sequence, show that $$5\mid u_m\Longleftrightarrow 5\mid m$$ Show: $\Longrightarrow$ In ...
0
votes
2answers
95 views

How to prove that the Fibonacci sequence $7\mid U_m\Longrightarrow 8\mid m$ and $4\mid U_m\Longrightarrow 6\mid m$

How to prove that the Fibonacci sequence $$7\mid U_m\Longrightarrow 8\mid m$$ and $$4\mid U_m\Longrightarrow 6\mid m$$I was confused because there $\{ 4,7 \}$ in Fibonacci sequece
4
votes
1answer
44 views

Closed form as sum and combinatorial of Fibonacci numbers

How can I prove that the Fibonacci numbers that are defined as $F_n=F_{n-1}+F_{n-2}, \; n \geq 2$ and $F_0=0,\ F_1=1,\ F_2=1$ have the form: $$F_n=\sum_{k=0}^{n-1} \binom{n-1-k}{k}, \; n\ge 2 $$ I ...
32
votes
5answers
645 views

Is ${F_{n}}^2 - 28$ always a composite number?

The problem: Prove or disprove that if ${F_{n}}$ is $n$-th Fibonacci number, and $n>5$ $${F_{n}}^2 - 28$$ cannot be a prime. I came to this accidentally while trying to solve another ...
10
votes
4answers
206 views

Linear Combinations of Fibonacci Numbers (integer coefficients)

While working on problem #2 on Project Euler, I came across the need to express $F_n$ as a linear combination of $F_{n-3}$ and $F_{n-6}$. This is relatively simple to do: $$\begin{align} F_n &= ...
0
votes
2answers
55 views

Help with a proof involving Fibonacci numbers.

I'm working through SICP MIT course, and I'm a little lost on how to prove the following statement. I think I'm able to demonstrate it, but have no idea how to prove this statement. I may ...
7
votes
3answers
481 views

Fibonacci numbers divisible by $9$

The $n$th Fibonacci number $F_n$ is defined as follows,$$F_1=F_2=1\mbox{ and } F_{n+2}=F_{n+1}+F_{n}\mbox{ for } n\geq 1$$ I want to know how many of the first $1000$ Fibonacci numbers are divisible ...
31
votes
6answers
5k views

Proof the Fibonacci numbers are not a polynomial.

I was asked a while ago to prove there is no polynomial $P$ in $\mathbb R$ such that $P(i)=f_i$ for all $i\geq0$. I tried to get a proof as slick as possible and here's what I got. Let ...
6
votes
1answer
270 views

Irrationality of reciprocal Fibonacci constant

I read that it was proved that reciprocal Fibonacci constant $$\sum_{n} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} ...
3
votes
0answers
131 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
5
votes
1answer
240 views

What is the sum of Fibonacci reciprocals?

How can I calculate $\sum\limits_{n=1}^{\infty}\frac{1}{F_n}$, where $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$? Empirically, the result is around $3.35988566$. Is there a "more mathematical way" to ...
2
votes
4answers
621 views

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms?

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms? I was watching scam-school on youtube the other day and this number trick just astonished me. Can someone please ...
1
vote
1answer
42 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = ...
5
votes
5answers
279 views

Fibonacci Proof

Prove that: $$F_1F_2+F_2F_3+F_3F_4+\cdots+F_{2n-1}F_{2n}=F_{2n}^2$$ I set it up so: $$F^2(2k) + F(2k+1)F(2k+2) = F^2(2k+2)$$ I've tried: $$F(2k)^2 + F(2k+1)*F(2k+2) = ...
3
votes
4answers
42 views

Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$

This is a question about Fibonacci sequences, a sequence in which the previous terms build up upon the current term (e.g. $F_1 + F_2 = F_3$ where $F_1 = F_2 = 1$). How would I go about proving ...
18
votes
1answer
378 views

Fibonacci-related sum

Related to this question Find a solution for f(1/x)+f(1+x)=x, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th Fibonacci number and ...
0
votes
1answer
33 views

are there infinitely many primes in Fibonacci sequence

There is one proof about infinitude of prime with following method, http://www.ams.org/mathscinet-getitem?mr=2271540 Also it is well know that any two consecutive Fibonacci numbers are mutually ...
43
votes
3answers
507 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Using a symbolic computation software (Mathematica), I got the following interesting results: $$ \begin{align} \sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{1+\frac{1}{1+n^2}}}} &= ...
2
votes
4answers
125 views

Prove that $p$ divides $F_{p-1}+F_{p+1}-1$ [duplicate]

Given the Fibonacci sequence $(F_n)$, defined by $F_0=0,F_1=1, F_{n+2}=F_{n+1}+F_n$, and $p$ an odd prime number, how to prove that $p$ divides $F_{p-1}+F_{p+1}-1$? Is induction a good idea here? ...
2
votes
2answers
67 views

Sum of squares of Fibonacci Numbers

$$ \sum_{i=0}^{n} (F_{2i+1})^2 = \;?$$ I know that sum of squares of first $n$ Fibonacci numbers is $F_{n} \times F_{n+1}$.
0
votes
2answers
40 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
1
vote
1answer
35 views

Counting the sequences of coin flips that end HH after $n$ flips (a more efficient method?)

I figured out that for any given $n$ the number of sequences of heads and tails that satisfy the condition that HH wasn't flipped consecutively until flips $n-1$ and $n$ is equal to the $(n-1)$th ...
15
votes
2answers
981 views

Find all integer solutions for the equation $|5x^2 - y^2| = 4$

In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation $$|5x^2 - y^2| = 4$$ occur when $x$ is a Fibonacci number and $y$ is a Lucas number. ...
1
vote
4answers
51 views

Fibonacci sequences and related series

Let $\{a_n\}$ be a sequence such that $a_1=a_2=1$ and $a_{n+1}=a_n+a_{n-1}$ for $n\geq 2$. Prove that $\displaystyle \sum_{n=1}^\infty \frac{1}{a_n}$ converges. My work: Let $b_n=\frac{1}{a_n}$. ...
0
votes
2answers
169 views

Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
1
vote
2answers
83 views

Divisibility of $987x^n − F_nx^{16} + F_{n−16}$

If $F_n$ is $n^{th}$ Fibonacci number, and polynomials $P_n(x)$ are defined as $987x^n − F_nx^{16} + F_{n−16}$, prove that for all $n ≥ 1$, $P_n(x)$ is divisible by $x^2−x−1$. This is from a ...
1
vote
2answers
36 views

Modulus of sum of sequence of Fibonacci numbers

What is the most efficient way to find the modulus of sum of sequence of fibinacci numbers. For example (F(N) + F(N + 1) + ... + F(M)) mod 1000000007.
19
votes
6answers
266 views

Why does every “fibonacci like” series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that ...
15
votes
5answers
3k views

Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?

About this Fibonacci sequence, is it true that the remainders when divided by three repeat along with the sequence like this: Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
0
votes
1answer
48 views

Proofs Related to the Fibonacci Sequence

I need to prove several proofs related to the Fibonacci sequence and I don't have the faintest clue how to do so. Please help! Given that the Fibonacci sequence is defined as $f_n = f_{n-1} + ...
6
votes
2answers
478 views

Fibonacci identity: $f_{n-1}f_{n+1} - f_{n}^2 = (-1)^n$

Consider this Fibonacci equation: $$f_{n+1}^2 - f_nf_{n+2}$$ The problem asked to write a program with given n, output the the result of this equation. I could use ...
2
votes
4answers
64 views

Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction

Where $n \in \mathbb{N}$ and $$ F_n = \begin{cases} 0 & \text{ if } n = 0 \\ 1 & \text{ if } n = 1 \\ F_{n-1} + F_{n-2} & \text{ if } n > 1 ...
0
votes
0answers
68 views

Hyper sum of Fibonacci numbers

Let $F(n)$ be the $n$-th Fibonacci number. That is, $F(n)$ satisfies $F(0)=0,F(1)=1,F(n)=F(n−1)+F(n−2) (if n≥2).$ Let $f_k(n)$ be the function such that $f_0(n) = F(n)$, $f_k(n) = ...