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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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
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3answers
591 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 ...
32
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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 ...
7
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1answer
328 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} ...
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0answers
150 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. ...
6
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1answer
265 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
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4answers
937 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
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1answer
70 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
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5answers
286 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
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4answers
46 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 ...
21
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1answer
420 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
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1answer
82 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 ...
2
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4answers
132 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? ...
3
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2answers
85 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 $F_{n} \times F_{n+1}$.
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2answers
72 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
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1answer
40 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 ...
1
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4answers
57 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
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2answers
227 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 ...
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2answers
88 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 ...
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2answers
106 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.
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6answers
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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
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5answers
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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, ...
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1answer
61 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
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2answers
492 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
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4answers
73 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 ...
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2answers
398 views

how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
3
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0answers
65 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
4
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5answers
98 views

Prove that $F(1) + F(3) + F(5) + … + F(2n-1) = F(2n)$

(These are Fibonacci numbers; $f(1) = 0$, $f(3) = 1$, $f(5) = 5$, etc.) I'm having trouble proving this with induction, I know how to prove the base case and present the induction hypothesis but I'm ...
1
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1answer
53 views

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. ...
0
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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
102 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.
0
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3answers
96 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 ...
2
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2answers
70 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 ...
6
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1answer
106 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
125 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, ${F_{n}}^2 + 41$ ...
6
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1answer
487 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 ...
2
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3answers
299 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. ...
2
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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 ...
9
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1answer
362 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 ...
2
votes
1answer
32 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
106 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
156 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 ...
7
votes
4answers
5k views

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 ...
5
votes
1answer
184 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 ...
4
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1answer
82 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 $F_k$ ...
7
votes
3answers
86 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 ...
1
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2answers
133 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 ...
1
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2answers
77 views

A series for Fibonacci numbers.

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