0
votes
0answers
25 views

Proof of a formula for generalized Fibonacci numbers

I have done the verification for $$U_rU_{n−1} − U_{r−1}U_n = (−1)^{r−1}U_{n−r}$$ I realized when I was doing for $n=k+1$, the expression $U_rU_k − U_{r−1}U_{k+1}$ would not equate to ...
1
vote
0answers
37 views

proving Fibonacci numbers using mathematical Induction?

Can anyone confirm whether my answer is correct, please. Let suppose we have the following fibonacci numbers as shown: $f(0) = 0, f(1) = 1$, and $f(n) = f(n-1) + f(n-2)$ for $n \geq 2$. Prove that ...
-1
votes
2answers
54 views

Mathematical induction used on Fibonacci Sequence

I have no clue how to go about doing this question using induction. It states that the Fibonacci sequence is defined as: F0 = 0 F1 = 1 Fn = Fn-2 + Fn-1 for n>=2 Let S(n) = Fo + F1 + F2 +...+ ...
0
votes
1answer
68 views

Fibonacci numeration system

Instead of binary or decimal, the Kingdom of Leutonia uses an unusual system to represent numbers, based on the Fibonacci sequence. The Fibonacci sequence $F_0,F_1,F_2,\dots$ is defined recursively ...
0
votes
1answer
38 views

Write out the Leutonian numbers that represent the first 12 positive integers.

How could I write out the leutonian numbers that represent the first 12 positive integers ? I have no idea how to start.
0
votes
0answers
72 views

Fibonacci sequence: Prove the formula $f_{2n+1}=f_{n+1}^2 + f_n^2$ [duplicate]

I can't seem to figure out this proof. I'm using weak induction and always get stuck during the inductive step. Prove for n > 0: $$f_{2n+1} = f_{n+1}^2 + f_n^2$$
1
vote
2answers
79 views

nth convolved Fibonacci numbers of order 6 modulo m

Problem: Find the coefficient of xk in (1−x−x2)-6 modulo m. Constraints: k≤264 m≤105, m can be a composite number. I have 10^5 such queries to process in 2 sec, so O(log k) for each query ...
1
vote
1answer
56 views

Fibonacci numbers identity - proof by induction

$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$ I have done the base step for $k=1$ and it works. I realize we need to prove for $k+1$, so: $$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$ Could ...
1
vote
2answers
59 views

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers. Progress: $F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not ...
1
vote
4answers
150 views

Determine which Fibonacci numbers are even

(a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture. (b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical ...
1
vote
1answer
70 views

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers. I have tried using the fact that $F_{n^k} \bmod F_n = 0, k=1,2,3,...$ but that doesn't get me anywhere. Thanks!
1
vote
4answers
47 views

Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...
0
votes
1answer
46 views

Understanding Recursive algorithm using FIB

I am studying for an exam, and I came across this question, I think I got the answer correct, just need some validation. ...
0
votes
2answers
66 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
72 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 ...
2
votes
4answers
140 views

For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$

For the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_n+F_{n+1}$, prove that $$\sum_{i=1}^n F_i= F_{n+2} - 1$$ for $n\ge 1$. I don't quite know how to approach this problem. Can someone help and ...
2
votes
4answers
111 views

Fibonacci proof question [closed]

Show that $$f_{n+1}f_{n-1}-f_n^2=(-1)^n$$ when $n$ is a positive integer and $f_n$ is the $n$th Fibonacci number.
2
votes
2answers
58 views

Why is $\sum_{k=0}^{n} f(n,k) = F_{n+2}$?

If $f(n,k)$ is the number of $k$ size subsets of $[ n ] = { 1 , \ldots , n }$ which do not contain a pair of consecutive numbers, how can I show that $\sum_{k=0}^{n} f(n,k) = F_{n+2}$? ($F_{n}$ is the ...
2
votes
4answers
121 views

Fibonacci sequence proof

Prove the following: $$f_3+f_6+...f_{3n}= \frac 12(f_{3n+2}-1) \\ $$ For $n \ge 2$ Well I got the basis out of the way, so now I need to use induction: So that $P(k) \rightarrow P(k+1)$ for some ...
0
votes
3answers
278 views

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 $F_n$: ...
0
votes
2answers
254 views

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 ...
2
votes
3answers
636 views

Strong inductive proof for this inequality using the Fibonacci sequence.

Problem I need to determine for what natural numbers is $2n < F_n$, where $F_n$ is the $n^{th}$ Fibonacci number determined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1}+F_{n-2}$. I then need to ...
2
votes
2answers
412 views

How do I prove Binet's Formula? [duplicate]

My initial prompt is as follows: For $F_{0}=1$, $F_{1}=1$, and for $n\geq 1$, $F_{n+1}=F_{n}+F_{n-1}$. Prove for all $n\in \mathbb{N}$: ...
1
vote
3answers
286 views

Proof of identity $F_m F_n + F_{m−1} F_{n−1} = F_{m+n−1}$ for Fibonacci numbers

I'm lost on where to start on this proof: Using the fact that $A^m A^n = A^{m+n}$ , prove the identity $F_m F_n + F_{m−1} F_{n−1} = F_{m+n−1}$ I want to use induction starting with n = 1, but would ...
3
votes
3answers
150 views

“Fat” sets of integers and Fibonacci numbers

Let us call a set of integers "fat" if each of its elements is at least as large as its cardinality. For example, the set $\{10,4,5\}$ is fat, $\{1,562,13,2\}$ is not. Define $f(n)$ to count the ...
7
votes
2answers
418 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 ...
2
votes
6answers
256 views

Fibonacci equality, proving it someway

$ F_{2n} = F_n(F_n+2F_{n-1}) $ $ F_n $ is a nth Fibonacci number. I tried by induction but i didn't get anywhere
5
votes
2answers
233 views

Two sums with Fibonacci numbers

Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$ Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ ...
0
votes
2answers
167 views

Count the number of paths in the Graph $P_3$. Provide a Proof by Induction using the Fibonacci sequence.

Consider the graph $P_3$ : $n_1$$ \rightarrow$ $n_2$$\rightarrow$ $n_3$$\rightarrow$ $n_4$ we count 6 paths of length k=1, namely: $n_1$ $\rightarrow$ $n_2$ $n_2$ $\rightarrow$ $n_3$ ...
1
vote
0answers
81 views

Lucas numbers theory confused [duplicate]

Possible Duplicate: Prove this formula for the Fibonacci Sequence How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$ How do I go about ...
8
votes
4answers
1k views

Proof by Induction: Alternating Sum of Fibonacci Numbers [duplicate]

Possible Duplicate: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer This is a homework question so I'm looking to just be nudged in ...
1
vote
3answers
377 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 ...
2
votes
1answer
185 views

An identity involving Lucas numbers

Let $L_n$ be the Lucas numbers, defined by $L_n = F_{n-1} + F_{n+1}$ where f is the Fibonacci numbers. How to prove that $L_{2n+1} = \displaystyle \sum_{k=0}^{\lfloor n + 1/2\rfloor}\frac{2n+1}{2n+1 ...
11
votes
8answers
1k views

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 $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$ ...