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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Confusion on unberstanding the proof of induction regarding Fibonacci numbers

I am trying to understand the proof that "For all $n\geq 2, F_n^2-F_{n+1}F_{n-1}=(-1)^{n-1}$.Where $F_n$ stands for the Fibonacci number at $n$. I got this proof from a book and here is the proof. ...
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2answers
40 views

Understanding Fibonacci Proof

I'm trying to show that $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2} ∀ ≥1$$ where $$F_k = F_{k-1} + F_{k-2}$$ with $$F_0 = F_1 = 1$$ Let P(n) = $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2}$$ Basic Step: ...
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1answer
97 views

What does $F^2_n$ mean?

In this Wikipedia entry on Cassini's identity, I saw this equation: $F_{n-1}F_{n+1}-F^2_n=(-1)^n$ $F^2_n$, what does that mean? Is it a summation signs for n to 2? I don't know what it means.
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3answers
83 views

Matrices, determinants, and applications to identities involving Fibonacci numbers

Preamble It is well known that since: $$ \begin{pmatrix} F_{n+1} \\ F_n \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix} \begin{pmatrix} F_n & F_{n-1} ...
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0answers
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Fibonacci, prove that $F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$ with induction [duplicate]

I need to prove by induction that: $$F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$$ I did the following: Check if the statement holds for $n=1$: $$1\cdot 3-(2)^2=(-1)^1$$ Check if the statement ...
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2answers
68 views

Seven expressions involving $F_n$ an $L_n$ that are always composite

Prove that if $F_n$ an $L_n$ are Fibonacci and Lucas numbers respectively, and $n>2$, then $$F_{n-2}\times F_{n-1}\times F_{n+1}\times F_{n+2}-15$$ $$F_{n-2}\times F_{n-1}\times ...
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0answers
43 views

Induction: Fibonacci / Lucas Numbers [duplicate]

From Andrews' Number Theory, Chapter 1, Section 1, Problem 15: Prove, by induction, that $F_{2n} = F_nL_n$ where $F_n$ denotes the nth Fibonacci number and $L_n$ denotes the nth Lucas ...
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1answer
104 views

Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
6
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1answer
120 views

Prove or disprove that ${F_{n}}^2 + 41$ is always a composite

The problem: Prove or disprove: If $F_{n}$ is the $n^{th}$ Fibonacci number then $${F_{n}}^2 + 41$$ is always a composite number. It looks that if $n$ is not multiple of 12, ${F_{n}}^2 + 41$ ...
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1answer
470 views

Flaw in induction proof that the Fibonacci sequence is bounded by $(5/3)^n$

The Fibonacci sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n \ge 2, a_{n+1} = a_n + a_{n-1}$. Thus the sequence begins $$1,1,2,3,5,8,13,21,...$$ Prove that for all $n \ge 1, a_n ...
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4answers
9k views

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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3answers
265 views

Induction proof $F(n)^2 = F(n-1)F(n+1)+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence [duplicate]

Prove that $F{_n}^2 = F_{n-1}F_{n+1}+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8 and so on. Initial case n = 2: $$F(2)=1*2+-1=1$$ It is true. ...
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0answers
90 views

Who First Considered This Generalization of the Fibonacci Numbers?

I am looking for the author who originally researched a generalization of the Fibonacci numbers, which Koshy, in Chapter 7 his book Fibonacci and Lucas Numbers with Application refers to as the ...
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5answers
3k views

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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5answers
245 views

Binomial Sum Related to Fibonacci: $\sum\binom{n-i}j\binom{n-j}i=F_{2n+1}$

How would I prove $$ \sum\limits_{\vphantom{\large A}i\,,\,j\ \geq\ 0}{n-i \choose j} {n-j \choose i} =F_{2n+1} $$ where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of ...
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1answer
357 views

Construction of generating function from identity

I am trying to solve identity involving binomials and Fibonacci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
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1answer
31 views

How can we be sure of periodicity by testing some terms?

A mod $n$ Fibonacci sequence is simply defined as the Fibonacci sequence, except all terms are in mod $n$. Now to determine periodicity, the worked solutions computed the first 20 or so terms and ...
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1answer
92 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
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2answers
125 views

Remarkable relation between Fibonacci numbers and its squares!

There is a remarkable relation between Fibonacci numbers and its squares: $F^{2}_{n} +F^{2}_{n+1}=F_{2n+1}$. I know how to prove it using $F_{n}=\frac{\sqrt{5}}{5}((\frac{1+\sqrt{5}}{2})^n ...
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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 ...
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1answer
157 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 ...
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1answer
76 views

Summation of a multiple series involving Fibonacci numbers

Compute the sum $$\sum_{a_{2015} = 0}^{\infty} \sum_{a_{2014} = 0}^{a_{2015}} \sum_{a_{2013} = 0}^{a_{2014}} \cdots \sum_{a_{1} = 0}^{a_2} \sum_{k=0}^{a_1} \frac{F_{k}}{2^{a_{2015}}} $$ where $F_k$ ...
6
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3answers
77 views

If $f_{n-1}^2=(f_n/2)^2+h^2$ then $n=6$

How can I prove that if $f_n$ is a term of the Fibonacci sequence divisible by $4$ and if $$f_{n-1}^2=(f_n/2)^2+h^2,$$ $h\in\Bbb Z^+$ then $n=6$? I know that since $\gcd(f_k,f_{k+1})=1$ for every ...
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2answers
117 views

Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number [duplicate]

If $F_n$ is the $n$-th fibonacci number, then prove that, $$F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$$ I tried the idea of using Pascal's triangle, but it seems to need some ...
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2answers
73 views

A series for Fibonacci numbers.

How can I prove The Fibonacci sequence is encoded in the number $1/89$ i.e. $( 1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034 \ldots)$
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2answers
45 views

Relation between series and equations

There is following quotes from wiki on Plastic number: The powers of the plastic number $A(n) = ρ^n$ satisfy the recurrence relation $A(n) = A(n − 2) + A(n − 3)$ for $n > 2$. And 2nd is that ...
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3answers
217 views

Inequality of the Fibonacci sequence and the golden ratio

How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. Any help is really appreciated
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2answers
54 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
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3answers
74 views

Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation

How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series representation (something neat similar to $\frac{1}{1-x}$)? This reminds me of the power series ...
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4answers
61 views

How to prove this equation by induction?

I am trying to prove this equation by mathematical induction $$f_{n+1}f_{n-1} = f_{n}^{2}+(-1)^n$$ is true where $f_{n} = $ the nth number in the Fibonacci sequence. I don't quite get how to do this ...
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2answers
35 views

Problem on deriving binet formula

I'm trying to understand binet formula. I got a good explanation here. Please look at the link. Everything just fine but one thing. It said that $A_n = A_{n-1} + A_{n-2}$, which is fibonacci. But why ...
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3answers
84 views

Showing convergence of recursive sequence $A_{n+1}=\frac 1 {1+A_n}$

Given : $\forall n\in\Bbb N,\quad A_{n+1} = \frac 1 {1+A_n}$ and $A_1 = 0$ Show the sequence converges and find its limit. Briefly what I did was to create two sub-sequences with an index ...
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1answer
59 views

Looking for $\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$

I am looking for the sum of the series: $$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$ where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the ...
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1answer
43 views

Prove that the set of solutions to $F_{n+2} = F_{n+1} + F_n$ is of dimension 2

I was playing with the Fibonacci sequence, willing to prove that $$ F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right) $$ I did the usual, ...
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235 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
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2answers
112 views

Limit of ratio of successive n-nacci numbers?

The n-nacci numbers are defined as $${}_nF_k = {}_nF_{k - 1} + {}_nF_{k - 2} + \cdots + {}_nF_{k - n + 1}$$ Now, it's pretty well-known that the limit of successive $2$-nacci numbers (i.e. the ...
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1answer
33 views

Discreet Math - Given n>= 5 how many times does fib(4) occur?

I have been trying to solve the below problem (and similar problems) but I have no clue how to tackle it. Can please help me tackle this particular problem, and how to attack similar problems? The ...
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1answer
98 views

Question on Proof that the Fibonacci Word is Sturmian

I am currently reading a text where it is proved that the infinite Fibonacci Word $u$ defined as the limit of the sequence $$ u_n = \varphi^n(0) $$ where the morphism is given by $\varphi(0) = 01, ...
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4answers
213 views

A sum involving Fibonacci numbers, $\sum_{k=1}^\infty F_k/k!$

Let $F_k$ be Fibonacci numbers. I am looking for a closed form of the sum $\sum_{k=1}^\infty F_k/k!$. I tried to use Wolfram Alpha, but it is not doing the sum Fibonacci[k]/k! , k=1 to infinity. ...
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4answers
604 views

Sum of cubes of first n fibonacci numbers

Let $\{f_k\}$ be the sequence of fibonacci numbers. It is well-known that $\sum_{k=1}^n f_k=f_{n+2}-1$ and $\sum_{k=1}^n f_k^2=f_n f_{n+1}$ . Is there a formula for $\sum_{k=1}^n f_k^3$ ?
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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
199 views

Sequence with denominators of products of consecutive Fibonacci numbers

I'm trying to figure out a way to solve the value of this: $$\frac{1}{1\times 2}-\frac{1}{2\times 3}+\frac{1}{3\times 5}-\frac{1}{5\times 8}+\frac{1}{8\times 13}-\dots$$ The only thing I can come up ...
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4answers
133 views

Prove that the Fibonacci recursion diverges

I have this sequence with $ n \in \mathbb{N} $ $ f(1) = f(2) = 1 $ and $ f(n) = f(n-1) + f(n-2) $ for $n \ge 3$ I think this sequence is bounded below and unbounded above. So it's clear that this ...
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1answer
39 views

Limit of Ratio of Two Generalized Fibonacci Sequences

I am hoping someone can help me determine the limit of two unique generalized Fibonacci sequences. Most everyone is familiar with the much talked about $\lim_{x\to \infty}$ ...
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2answers
55 views

fibonnaci and lucas series technique

Well i have the following two problems involving fibonnaci sequences and lucas numbers, i know that they share the same technique, but i don't have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 ...
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2answers
32 views

Prove by induction fibonacci variation

Prove by induction: The fibonacci sequence is defined as follows: $f_1 = 1$, $f_2 = 1$ and $f_{n+2} = f_n + f_{n+1}$ for $n \geq 1$ Prove by induction that $f_1^2 + f_2^2 + \dotsb + f_n^2 = f_n ...
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1answer
71 views

New Identities for Generalized Fibonacci Numbers?

Over the past few months I have been investigating one the generalizations of the Fibonacci numbers, called the Generalized Fibonacci Numbers (GFNs). The GFNs are just like the regular Fibonacci ...
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7answers
1k views

How are we able to calculate specific numbers in the Fibonacci Sequence?

I was reading up on the Fibonacci Sequence when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly ...
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1answer
65 views

Induction proof for Fibonacci numbers

I am trying to get a hang on the induction method for proof, but I'm still dubious of many aspect of this proof regarding its application to sequences of integers, such as the Fibonacci sequence. ...
2
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1answer
42 views

What does “The closure of the shift-orbit of the Fibonacci word” mean?

Im trying to translate an article about rauzy fractal. But since my English is not good enough I cant understand this paragraph: ...