# Tagged Questions

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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### How to prove $\sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
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### Is this Bertrand's postulate-related statement valid?

Bertrand's postulate says: For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$. Is the following statement: For every $n>3$ there is always at least one ...
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### Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

So, $$1,1,2,3,5,8,13,21...$$ Any connection to primes?...it appears not. However, in between the Fibonacci numbers are how much primes? Let's see: 1 and 1 ZERO 1 and 2 NADA 2 and 3 ZILCH 3 and 5 ZIP ...
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### How to solve the difference equation $u_n = u_{n-1} + u_{n-2}+1$

Given that: $$$$u_n=\begin{cases} 1, & \text{if 0\leq n\leq1}\\ u_{n-1} + u_{n-2}+1, & \text{if n>1} \end{cases}$$$$ How do you solve this ...
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### Proving $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ for $n \in \mathbb{N}_{>1}$ when $F_n$ is the nth Fibonacci number

Let $F_n$ be defined as the nth Fibonacci number. Prove that $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ with $n \in \mathbb{N}_{>1}$ My approach thus far was to use induction over $n$. ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\{r_1,r_2,r_3,\dots, r_n \}$. $$\displaystyle P(x) = ... 1answer 68 views ### Covering a rectangle of size n\times1 with dominos A rectangle of size n\times1 is given. (a) In how many ways the rectangle can be covered with dominoes of size 1\times1 and 2\times1? (b) In how many ways the rectangle can be covered with ... 0answers 76 views ### Show that for a given s there are a finite number of Fibonacci number of form n^2+s It is well known that the last Fibonacci number F_k such that \exists \ n \in \Bbb{N} : F_k = n^2 is 144. Thus there are only 4 perfect squares among the Fibonacci sequence (assuming you ... 4answers 76 views ### Why the GCD of any two consecutive fibonnaci numbers is 1? Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof. I did the following:$$F_n=F_{n-1}+F_{n-2} \\ ...
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 ...