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 sequences

I have the following: $$ f_3+f_6 + \dots+f_{3n} = \frac 12 (f_{3n+2}-1) $$ for $f_0=0$ and $f_1=1$ When I calculate $n\ge2$ and $f_n= f_{n-1}+f_{n-2}$, I get: LHS = 8 while RHS = 10. LHS $$f_6 ...
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Exponential lower bound for Fibonacci numbers

Can someone show me how to solve through induction, $F(N) \geq (3/2)^N$ for all $N\geq N_0$, where $F(n)$ is the Fibonacci function and $N_0$ is some positive integer. I know that $N_0$ should be ...
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Simplifying Sigma Notation

I am working on the proof on showing the ratio of two consecutive Fibonacci numbers converges to the golden ratio to explain to a student I am tutoring. I am getting to some confusion in a ...
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Fibonacci combinatorial identity

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

I have a bit difficulty to proofe that two consecutive numbers are coprime. I have the following The property $P(n)$ is the equation $(F_{n+1},F_n)=1$ where F_i the sequence of fibonacci is and $n ...
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Fibonacci Proof: Prove that $\frac{F_n-F_{n+16}}{7}$ is always an odd integer.

Im stuck on this question, i have to prove that $$\frac{F_n-F_{n+16}}{7}$$ is always an odd integer. I tried induction to do this but i just can't see how to prove it. thanks for any help
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Induction proof with Fibonacci numbers

Prove by induction that for Fibonacci numbers from some index $i > 10$ $1.5^i ≤ f_i ≤ 2^i$ Notice! Because Fibonacci number is a sum of 2 previous Fibonacci numbers, in the induction hypothesis ...
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How do I use telescopic cancellation to prove the Fibonacci Sum Identity

I am reading a textbook which attempts to prove the Fibonacci Sum Identity by rearranging the Fibonacci recurrence relation as follows and then using telescoping cancellation to prove the identity: ...
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Matrix powers and recurrence relations

The nth Fibonacci number can be found by raising the matrix $\begin{pmatrix}1 & 1 \\ 1 & 0 \end{pmatrix}$ to the nth power. Are there other recurrence formulas that can be solved like this? ...
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how would you represent the tribonacci spiral?

As we all know, the Fibonacci Sequence has a very famous spiral representation. Here we can see it. Now it's easy to see the fibonacci is the sum of the 2 squares that compose it to form a ...
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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 ...
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how to find nth term in a fibonacci series or sum of a series of fibonacci numbers

A series is given as $1,6,7,13,20,33,.......$ and so on Find the sum of first 52 terms? what i know is The sum of the first n Fibonacci numbers is the [(n + 2)nd Fibonacci number - 1] . So the sum ...
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Number of Fibonacci numbers in a range

The definition of the Fibonacci numbers is given by: $$\begin{align}f_1 &= 1;\\ f_2 &= 2;\\ f_n &= f_{n-1} + f_{n-2},\qquad (n >= 3); \end{align}$$ now we are given two numbers $a$ ...
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Gcd of every other Fibonacci number

Let $f_n$ be Fibonacci Sequence. $$gcd(f_{n},f_{n+2})=1,\quad \forall\,n\in\mathbb{N}.$$ Prove Could you help me with this one? I have done the base case, I just can't figure out the inductive step. ...
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N-binacci numbers and ratios generated by them

I was bored at home today and playing with "n-bonacci" numbers, numbers generated by $$x_0=0,x_1=1,...,x_k=k; x_n=\sum_{i=0}^{n-1}x_i$$ I made an assumption based upon the quadratic responsible for ...
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Generating Function for the adjusted Fibonacci numbers

The task is to find another relation for the adjusted Fibonacci numbers. I've found there genertaing function $$A(x)=\dfrac{1}{1-x-x^{2}}$$ Furthermore I've created the generating function in a ...
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A Non-recursive Fibonacci Sequence

How can I determine the general term of a Fibonacci Sequence? Is it possible for any one to calculate F2013 and large numbers like this? Is there a general formula for the nth term of the Fibonacci ...
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Prove the Number of Additions of Fibonacci Number Algorithm

I am studying for a final exam and I'm having trouble with this question: The following recursive algorithm FIB takes as input an integer $n \ge 0$ and returns the $n$-th Fibonacci number $F_n$: ...
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83 views

Prove the following identity for Fibonacci numbers

Prove this: for any positive integer $a,b,c$, $F_{a+b+c+3}=F_{a+2}(F_{b+2}F_{c+1}+F_{b+1}F_c)+F_{a+1}(F_{b+1}F_{c+1}+F_bF_c)$ Is there any way other than induction to prove this?
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GCD of Fibonacci-like recurrence relation

What is the greatest common denominator of $t(c^a)$ and $t(c^b)$, if $t(n) := k_1 f_1^n + k_2 f_2^n $? I already found out that the gcd is always a member of $t(n), n \in N $. $t(n)$ was originally ...
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Finite bit strings that do not contain '$00$'

I am studying for an exam and I am having trouble with this practice question: In this question, we consider finite bit strings that do not contain $00$. Examples of such bitstrings are $0101010101$ ...
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Help with Induction proof on Fibonacci sequence?

I can't seem to solve this problem. It is: The Fibonacci numbers $F(0), F(1), F(2),\dots $ are defined as follows: $F(0) ::= 0$ $F(1) ::= 1$ $F(n) ::= F(n-1) + F(n-2)\qquad(\forall n \ge 2 $) ...
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Proving Lucas Identity by Induction

I am trying to prove the following identity (I decided to use induction, but if that's not the best way feel free to mention that in the answers): $$L^2_n = 5F^2_n + 4(-1)^n \space\space ...
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Fibonacci Sequence Exercise

I need some help checking the following solution. The Fib sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n\geq 2$, $a_{n+1} = a_n + a_{n-1}$. Thus, the sequence begins: 1, 1, 2, 3, 5, 8, ...
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Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
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find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
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Calculating Pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the Pisano period of a number. After I researched my way through the web, I found ...
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fibonacci numbers mod some prime number

Moderator Note: This is a current Code Chef challenge question. When the current challenge ends on 15 October 2013 this question will be unlocked. I have prime numbers ($\geq11$) and of the form ...
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Cycles in the Generalized Fibonacci Sequence modulo a Prime

Suppose I have a fibonacci sequence 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 Now if I have a modulo 5 fibonacci sequence,it will look like ...
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Probability of Runs of Heads of Length N [duplicate]

For example: $“THHTHTTHHHTHTHTTHHTHT”$ contains 1 run of heads of length 3, 2 runs of length 2, and 4 runs of length 1. Assuming $P(H) = p$ and $P(T) = (1-p)$, calculate (using properties such as ...
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Fibonacci with Mortal Bunnies

I am trying to understand a twist on the Fibonacci bunnies scenario, where the bunnies die x generations after their birth (where x is a positive integer). An example is shown here. I understand the ...
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Divisibility of Fibonacci numbers

This question is inspired by a Project Euler problem I was working on. Noticing something that did not make sense led me to the conclusion that for all primes $p$ ending in $1$ or $9$, the $(p-1)$st ...
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Showing the fibonacci sequence for any number n

My lecturer was explaining how the Fibonacci sequence can be displayed for a number n. The formula is fib(n)=fib(n-1) +fib(n-2) ...
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Using induction to prove a result about the Fibonacci sequence

The Fibonacci sequence $F_0, F_1, F_2,...,$ are defined by the rule $$F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$$ Use induction to prove that $F_n\geq2^{0.5n}$ for $n\geq 6$ So far I have done the ...
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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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Prove the given property of the Fibonacci numbers directly

The definition of the Fibonacci numbers is as follows: $F(0)=0$, $F(1)=1$, $F(n)=F(n-2)+F(n-1)$ for $n ≥ 2$. Prove the given property of the Fibonacci numbers directly from the definition (hint: do ...
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Determine the number of n-term sequences of 0s and 1s containing no two consecutive $0$s

I am reading a chapter about Fibonacci number and generating function. And there's a question come up but without solution. I think about it for quite some time, but still can't come up with a ...
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Fibonacci Mystery

I saw this on a "numberphile" video and tried to prove it but couldn't do anything. Theorem: Let $n \ge 2$ and $F_m$ is the $m^{\text{th}}$ number in the Fibonacci sequence. Then, if we look all ...
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Please help to understand Fibonacci numbers' property.

Theorem: The Fibonacci numbers are defined recursively thus: $$x_{n+1} = x_n + x_{n-1}$$ with $$x_1=x_2=1.$$ Prove that $$x_n=(a^n-b^n)/(a-b),$$ where $a$ and $b$ are the roots of the quadratic ...
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Counting function for Fibonacci numbers

Are there some results about "Fibonacci-counting function" - the function counting the number of Fibonacci numbers less than or equal to some real number x?
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Fibonacci sequense, problem od division

How to show that $7\mid F_m\Longrightarrow 8\mid m$ and $4\mid F_m\Longrightarrow 6\mid m$, knowing that (I) Two consecutive terms in the Fibonacci sequence are relatively prime. (II) In ...
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Strong Induction Proof: Fibonacci number even if and only if 3 divides index

The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$ Proof by Strong Induction : ...
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Proof by induction on Fibonacci numbers: show that $f_n\mid f_{2n}$

I was studying Mathematical Induction when I came across the following problem: The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation- $f_n = f_{n-1} + ...
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The Principle of Mathematical Induction

The question is Let $( F_0, F_1, F_2,... )$ be the Fibonacci sequence defined by $F_0=0,\, F_1=1, and F_{n+1}=F_n+F_{n-1}$, n greater than or equal to 1. Prove the following identities. ...
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The relation between piano 12-scale and Fibonacci?

One of my books says there is a relation between the chromatic musical scale [CC#DD#EFF#GG#AA#BC] and the Fibonacci sequence. So...what's the relation?
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Sum of Fibonacci-Numbers

Is there a closed formula for the sum of the first $n$ even (or odd) Fibonacci numbers, like there is one for the $n$th number (Moivre-Binet)?
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Closed formula for the Fibonacci sequence (induction) [closed]

Let $f (n)$ denote the $n^{th}$ term of the sequence of integers given by the equation $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1 \land f(2) = 1$, then using principle of mathematical ...
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Monotonicity of the sequence $ ( F_n^{\frac{1}{n}} ) $, where $ ( F_n ) $ is the Fibonacci sequence

Let $ F_n = F_{n-1} + F_{n-2} $ with $ F_0 = 1 $, $ F_1 = 1 $ (the Fibonacci sequence). I would like to know whether $ F_n^{\frac{1}{n}} $ is monotonically increasing in $ n $. It is not difficult to ...
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Fibonacci using proof by induction: $\sum_{i=1}^{n-2}F_i=F_n-2$

everyone. I have been assigned an induction problem which requires me to use induction with the Fibonacci sequence. The summation states: $$\sum_{i=1}^{n-2}F_i=F_n-2\;,$$ with $F_0=F_1=1$. I ...
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Why is the reciprocal of the second Fibonacci number negative?

The second Fibonacci number is 1, so it's reciprocal should be 1, right? Why is it that I get $-1$ when I plug in $2$ for n in the reciprocal of Binet's equation ...