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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Fibonacci modular results

Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which ...
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4answers
7k views

The generating function for the Fibonacci numbers

Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$.
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8answers
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Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$

Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$ ...
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4answers
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Prove this formula for the Fibonacci Sequence

This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$. Show that ...
7
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3answers
690 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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5answers
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How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
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3answers
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Fibonacci sequence divisible by 7?

Make and prove a conjecture about when the Fibonacci sequence, $F_n$, is divisible by $7$. I've realized it's when $n$ is a multiple of $8$. I just don't know how to go about proving it.
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2answers
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Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
2
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1answer
719 views

Causal Inverse Z-Transform of Fibonacci

Say the Fibonacci sequence is defined by: $y(n) = y(n-1) + y(n-2)$ initial conditions: $y(0)=0, y(1)=1$ I incorporate those initial conditions as: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ ...
17
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1answer
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Prove that two any consecutive terms of Fibonacci sequence are relatively prime

Prove that two any consecutive terms of Fibonacci sequence are relatively prime My attempt: We have $f_1 = 1, f_2 = 1, f_3 = 2...$. So obviously $\gcd(f1, f2) = 1$. Suppose that $\gcd(f_n, ...
8
votes
3answers
987 views

Prove by induction Fibonacci equality

[question:] Prove by induction that the i th Fibonacci number satisfies the equality $$F_i=\frac{\phi^i-\hat{\phi^i}}{\sqrt5}$$where $\phi$ is the golden ratio and $\hat{\phi}$ is its conjugate. ...
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1answer
905 views

Recurrence relation, Fibonacci numbers

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$. $(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ ...
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7answers
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Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$ [duplicate]

I need to prove that there is the following equality: $$ \sum\limits_{k=0}^n {n-k \choose k} = F_{n} $$ where $F_{n}$ is a n-th Fibonacci number. The problem seems easy but I can't find the way to ...
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2answers
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Fibonacci, tribonacci and other similar sequences

I know the sequence called the Fibonacci sequence; it's defined like: $\begin{align*} F_0&=0\\ F_1&=1\\ F_2&=F_0+F_1\\ &\vdots\\ Fn&=F_{n-1} + F_{n-2}\end{align*}$ And we ...
4
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4answers
958 views

How to prove this is a rational number

I'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
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3answers
478 views

Significance of starting the Fibonacci sequence with 0, 1…

DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic. QUESTION: What is the significance of ...
4
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1answer
857 views

Sum of product of Fibonacci numbers

I want to calculate the sum of product of Fibonacci number for a given $n$. That is, for given $n$, say $$F_1 F_n + F_2 F_{n-1} + F_3 F_{n-2} + F_4 F_{n-3} + F_5 F_{n-4} + \cdots.$$ what should be ...
2
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2answers
218 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: ...
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4answers
444 views

Divisibility and the Fibonacci sequence

While studying the Fibonacci sequence I encountered this problem in the handout, and I can not understand how to do it. Show that if the Fibonacci sequence has a term divisible by a natural number ...
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2answers
907 views

Summation of series of product of Fibonacci numbers

What is the sum of following product of Fibonacci numbers $$\sum_{k=1}^{n-1} Fib(k)*Fib(n+3-k)$$ can anyone suggest only approach to find general term?
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5answers
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Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
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5answers
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Checking if a number is a Fibonacci or not?

The standard way (other than generating up to $N$) is to check if $(5N^2 + 4)$ or $(5N^2 - 4)$ is a perfect square. What is the mathematical logic behind this? Also, is there any other way for ...
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3answers
541 views

Proof of this result related to Fibonacci numbers?

$$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$$ Somebody has any idea how to go about proving this result? I know a proof by ...
17
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1answer
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Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it ...
16
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2answers
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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. ...
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3answers
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Prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
7
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1answer
722 views

Another way to go about proving the limit of Fibonacci's sequence quotient.

It is not difficult to inductively prove that $$\eqalign{ & \phi = \phi + 0 \cr & {\phi ^2} = \phi + 1 \cr & {\phi ^3} = 2\phi + 1 \cr & {\phi ^4} = 3\phi + 2 ...
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3answers
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For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$

The definition of a Fibonacci number is as follows: $$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$ Prove the given property of the Fibonacci numbers for all n greater than or equal to 1. ...
5
votes
2answers
206 views

Number of bitstrings with $000$ as substring

I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$? Now there are many ways to go about this but if I choose starting ...
54
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4answers
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Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
36
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5answers
730 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 $n>5$, than $${F_{n}}^2 - 28$$ cannot be a prime. I came to this accidentally while trying to solve ...
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15answers
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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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1answer
607 views

How to prove that Fibonacci number is integer?

How to prove that formula for Fibonacci numbers are always integers, for all $n$: $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - ...
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1answer
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Induction Proof: Formula for Sum of n Fibonacci Numbers

I am stuck though on the way to prove this statement of fibonacci numbers by induction : my steps: definition: $F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$ The Hypothesis is: $\sum_{i=0}^{n} ...
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4answers
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Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$

I can't seem to solve this problem. It is: The Fibonacci numbers $F(0), F(1), F(2),\dots $ are defined as follows: \begin{align} F(0) &::= 0 \\ F(1) &::= 1 \\ F(n) &::= F(n-1) + ...
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2answers
215 views

Another Fibonacci identity

Here's a problem that is leading me in circles. Consider the Fibonacci number $F_n$ defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 2$. Prove that $F_{2n-1} = F_{n}^2 ...
3
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Another way to go about proving Binet's Formula

As I showed in another question of mine, it is easy to prove that $$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$ given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$. Now, extending $(1)$ ...
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4answers
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Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares

Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step. Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$ I ...
14
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1answer
513 views

Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?

$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$ (Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ \ldots )$ This is ...
12
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2answers
315 views

Interpolated Fibonacci numbers - real or complex?

The common Binet-formula for the Fibonacci-numbers $$ f_n = {\varphi^n- (1-\varphi)^n \over \sqrt 5 } \small {\qquad \qquad \text{ where }\varphi={1+\sqrt 5\over 2}}$$ allows interpolation to ...
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3answers
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Infinite Series: Fibonacci/ $2^n$

I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner) In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... each term ...
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3answers
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How many numbers are in the Fibonacci sequence

Assuming I'm asked to generate Fibonacci numbers up to N, how many numbers will I generate? I'm looking for the count of Fibonacci numbers up to N, not the Nth number. So, as an example, if I ...
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5answers
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Why does the Fibonacci Series start with 0, 1?

The Fibonacci Series is based on the principle that the succeeding number is the sum of the previous two numbers. Then how is it logical to start with a 0? Shouldn't it start with 1 directly?
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2answers
167 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,\ \ ...
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1answer
143 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as: Lets say we have the number $p$ which is an odd prime. Prove that: $F_{p-1} + F_{p+1} -1$ Is divisible by $p$. Prove that for any given $n$ real ...
5
votes
3answers
443 views

Closed form for the sum of even fibonacci numbers?

I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for $$ \sum_{i=1}^n F_{3i} $$ ...
14
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2answers
573 views

Fibonacci[n]-1 is always composite for n>6. why?

In[11]:= Select[Table[Fibonacci[n], {n, 1, 10000}], PrimeQ[# - 1] &] Out[11]= {3, 8} Edit: Fibonacci[n]-1 is always composite for n>6. why? $$\sum\limits_{i = 0}^n {{F_i}} = {F_{n + 2}} - 1$$ ...
10
votes
3answers
261 views

Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$

Evaluate the sum: $$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$ where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.
8
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1answer
352 views

Prove: the intersection of Fibonacci sequence and Mersenne sequence is just $\{1,3\}$

$$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$$ Here $\varphi = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?
3
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2answers
133 views

Fibonacci combinatorial identity

Can someone explain how to prove the following identity involving Fibonacci sequence $F_n$ $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ?