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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4
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2answers
363 views

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 ...
5
votes
3answers
1k views

Fibonacci trick and proving it. [duplicate]

I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is: ...
3
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1answer
98 views

Question about generating function of kind of fibonacci partial sum

$F_n$ here is $n$-th fibonacci number We know that $$\sum_{n=0}^\infty \left(\sum_{k=0}^n F_kF_{n-k}\right)x^n$$ is a generating function of multiplying two G.F: $a_n =\langle F_n \rangle$ and $b_n = ...
0
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2answers
73 views

Interesting a Fibonacci quesiton. Need help.

Alice claims that she knows another formula for the Fibonacci numbers: Fn = $e^{n/2−1}$ for $n = 1,2,\cdots$ (where $e = 2.718281828$... is, naturally, the base of the natural logarithm). Is she ...
2
votes
1answer
439 views

Help with induction proof for formula connecting Pascal's Triangle with Fibonacci Numbers

I am in the middle of writing my own math's paper on the topic of Pascal's Triangle. During the investigation I have came up with a formula for counting elements of Fibonacci Sequence using the ...
2
votes
0answers
65 views

Summation involving Fibonacci numbers

Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.
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1answer
67 views

Fibonacci Proof with Induction [duplicate]

$$f(n) = \left\{\begin{matrix} 0 & n=1\\ 1 & n=2\\ f_{n-1} + f_{n-2} & n\geqslant 2\end{matrix}\right.$$ How can I prove by induction that $$f_{n} \geq \left ( 1.5 \right )^{n-1}$$ ...
2
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2answers
92 views

Fibonacci Proof Using Induction

$$f(n) = \left\{\begin{matrix} 0 & n=1\\ 1 & n=2\\ f_{n-1} + f_{n-2} & n\geqslant 2\end{matrix}\right.$$ How can I prove by induction that $$f_{n} \leq \left ( \frac{1+\sqrt{5}}{2} ...
2
votes
1answer
144 views

Fibonacci proof by induction

I have fibonacci numbers defined as such: $$ f(n) = f(n-1) + f(n-2)$$with$$ f(0) = 0 $$$$f(1) = 1$$ I have to prove that $$ F(n) \geq 1.5^{n-1}, n \geq 6$$ Base Case: $$ f(6) = 8 \geq (1.5)^5 ...
3
votes
1answer
113 views

A Fibonacci series

Let $F_n$ be the $n^{th}$ term of the Fibonacci sequence. That is, $F_1 = F_2 = 1$ and $F_n$ is defined recursively for $n\geq3$ by $F_n = F_{n-2}+F_{n-1}$. It is a known fact that $$ ...
0
votes
1answer
49 views

how to use Newton's polynomials calculate this?

Say we have Fibonacci recurrence: $ F_{n+2}=F_{n+1}+F_n$, with $F_0=1,F_1=1$ We can write $F_n = a \alpha^n + b \beta ^n$, so how do we use Newton's polynomials to determine the value of $\alpha^r + ...
3
votes
2answers
115 views

a manipulation of Fibonacci recurrence

Let $F_n$ be the Fibonacci number, and we know $F_{n+2} = F_{n+1} + F_{n} $ with $F_0 =1,F_1 = 1$ And this can be manipulated to $F_{n+6} = 4F_{n+3} + F_n$ if we let n be a multiple of 3, we can ...
2
votes
2answers
147 views

Sum $\frac{1}{1\times2}+\frac{1}{1\times3}+\frac{1}{2\times5}+\frac{1}{3\times8}+\cdots$

If $f_n$ is the Fibonacci series, with $1,1,2,3,5,8,\ldots$ prove that $$\sum_{i=2}^\infty\frac{1}{f_{i-1}\cdot f_{i+1}} = 1$$ So my idea was to try to convert this series into a telescoping sum ...
1
vote
3answers
91 views

Prove the given property of the Fibonacci numbers

I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular: For all $n \in \mathbb N$, $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. I ...
4
votes
1answer
254 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
11
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4answers
5k views

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 ...
2
votes
2answers
128 views

What is the relation between this binary number with no two 1 side by side and fibonacci sequence?

I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side. Now as an example ...
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2answers
280 views

Generating function for squared fibonacci numbers

We know that generating function for fibonacci numbers is $$B(x)=\frac{x}{1-x-x^2}$$ How can we calculate $B(x)^2$? I thought that, if we have $B(x)=F_n*x^n$ then $$B(x)*B(x) = \sum_{n=0}^\infty ...
1
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1answer
58 views

Generating function for kind of sum of Fibonacci numbers

Let's have a sequence $$a_n = \sum_{i=0}^n F_iF_{n-i}$$ where $F_n$ is n-th Fibonacci number. I tried to solve it somehow, but i'm pretty stuck. Defining Fibonacci numbers $$b_0=0, b_1=1, ...
1
vote
2answers
130 views

What is the asymptotical bound of this recurrence relation?

I have the recurrence relation, with two initial conditions $$T(n) = T(n-1) + T(n-2) + O(1)$$ $$T(0) = 1, \qquad T(1) = 1$$ With the help of Wolfram Alpha, I managed to get the result of ...
1
vote
1answer
48 views

How do I calculate the number of members in a limited Fibonacci series? [duplicate]

Looking for an algorithm that will give me the number of members that will result from calculating a Fibonacci series, given a particular limit. For example, if I start the series at 1 and limit my ...
0
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1answer
78 views

Generalisation of Fibonacci

Somehow a generalisation of the fibonacci numbers, do numbers created by the formula $ F(n) = F(n-1) + [F(n-1)-F(n-2)+F(n-3)-F(n-4)+F(n-5)-F(n-6).....]$ with $F(1) = 1$ have a specific name?
0
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2answers
78 views

Fibonacci Sequence and odd/even addition [duplicate]

Prove that f0 – f1 + f2 - … - f2n-1 + f2n = f2n-1 – 1. For n is all positive numbers. I have an idea to what I must do, but I can't figure what the base case is. I think it is f(0) = 0 and f(1) = 1. ...
2
votes
1answer
122 views

No advantage to the closed form for Fibonacci numbers?

The closed forms for the Fibonacci sequence, such as: $$F_n=\frac{\varphi^n-\widehat\varphi^n}{\sqrt5}=\frac{\varphi^n}{\sqrt5}-\frac{\widehat\varphi^n}{\sqrt5}\;,$$ the Binet formula, do not seem ...
6
votes
4answers
108 views

Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research ...
3
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4answers
6k views

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 ...
1
vote
5answers
503 views

Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]

We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}. $$ I wrote a script and found that this ...
2
votes
3answers
467 views

Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer

Letting $f_n$ be the Fibonacci numbers, show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer. Just some homework help. Need to prove. Thank you in ...
6
votes
3answers
699 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 $$ \sum_{i=1}^n F_{3i} $$ ...
18
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1answer
2k views

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 ...
6
votes
4answers
852 views

Fibonacci identity proof

I've been struggled for this identity for a while, how can I use combinatorial proof to prove the Fibonacci identity $$F_2+F_5+\dots+F_{3n-1}=\frac{F_{3n+1}-1}{2}$$ I know that $F_n$ is number of ...
4
votes
3answers
114 views

Write $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, where $F_n$ is the n-th element of the Fibonacci sequence?

The exercise asks to express the following: $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, not necessarily a closed one. The previous problem in the set was the same, with a different ...
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2answers
134 views

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)?
0
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2answers
157 views

Prove that a Fibonacci number is greater than $ φ^n$

How can I prove the following: If $f_n$ is a number of the Fibonacci sequence and φ= $\frac{1+\sqrt{5}}2$, then $f_n > φ^n$ for every $n >2$? I have tried using induction but I can't ...
1
vote
3answers
321 views

partial Fibonacci summation

Let $F_{n}$ be the n-th Fibonacci number. How to calculate the summation like following: $\sum_{n \geq 0} F_{3n} \cdot 2^{-3n}$
3
votes
1answer
65 views

All pairs sum to a different value

If we consider the integers $\{1,\dots,n\}$, what is the size of the largest subset $A$ so that all distinct pairs $x, y \in A$ sum to a different value? For this to make sense $(x,y)$ and $(y,x)$ ...
4
votes
2answers
951 views

Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]

Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove ...
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1answer
261 views

Calculate Number of ways to make the grid

We wish to tile a grid of size Nx2 with rectangles (dominoes) of 2x1 (in either orientation).For given N I need to find the number of different ways to tile the grid. EXAMPLE : For N=1 answer is 1 ...
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0answers
59 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
5
votes
5answers
1k views

Where do the first two numbers of Fibonacci Sequence come from? [duplicate]

I'm trying to code a simple algorithm that prints out the $n^{th}$ Fibonacci number. However, my program requires the initial seed values $F_0 = 0$ and $F_1 = 1$, when I'm hopeful I can figure ...
0
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1answer
47 views

asymptotics of $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $

Starting from $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $ and letting $F_k \approx \phi^k$, I am hoping to find the corresponding statement for the Golden ratio: $\phi^n = 2 ...
2
votes
3answers
3k views

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
6
votes
1answer
644 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 - ...
5
votes
4answers
2k 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? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
5
votes
2answers
223 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 ...
5
votes
1answer
495 views

Sum of digits in Fibonacci sequence

This is my first question here so please go easy on me. If you add the digits of each number on the Fibonacci sequence until your number is less than 10, it seems that you get a pattern of 24 numbers ...
1
vote
1answer
1k views

Formula for binary sequences of length m with no n consecutive 1s?

Formula for binary sequences of length $m$ with no $n$ consecutive $1$s? I know The number of binary strings of length $m$ without consecutive $1$s is the Fibonacci number $F_{m+2}$. But how about ...
8
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1answer
173 views

Is this number rational or irrational?

Start writing down the Fibonacci numbers, using two digits for each one 01 01 02 03 05 08 13 21 34 55 ... Eventually you will reach three digit numbers. When ...
2
votes
2answers
118 views

Internet problem solving contest question

I am trying to solve a problem from the IPSC http://ipsc.ksp.sk/2001/real/problems/f.html It basically asks to compute the following recursion. ...
3
votes
3answers
368 views

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 $(1)$ ...