# 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
### 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 $β=\dfrac{1-\sqrt{5}}{2}.$ ...
### 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} = ...