# fibonacci

1. ### Fibonacci relative prime proof

Hey Guys, learning discrete math by myself and while I was working with a proof, I got stuck, I checked the solution and I understand the big picture but I can't seem to understand the arithmetic, here's how it goes: By induction Let P(n) be the proposition that Fn and Fn+1 are relatively...
2. ### Fibonacci Seq bit string proof. Show that pattern can't have three consecutive 1s

I'm having a little trouble figuring out how to prove this. So, the sequence is as follows: S0 = 0 S1 = 1 S(n-2)S(n-1) for all n >= 2 That means S2 = 10 S3 = 101 S4 = 10110 S5 = 10110101 Now the question that I'm stuck on is: Prove that the pattern can't have three consecutive ones...
3. ### Fibonacci's Cubic

'Sup guys. I'm new here but I'm REALLY having a hard time in my math history class. The instructor is a joke tbh. But enough about that. I need help with a couple of proofs and I've exhausted my resources. 1. The problem states: "Consider Fibonacci's Cubic x3+2x2+10x=20 a) Show that this has...
4. ### Power Series for Fibonacci Numbers

Statement of problem: Let a_{n} denote the nth Fibonacci number, defined recursively by a_{0}=a_{1}=1, a_{n+1}=a_{n}+a_{n-1}. Let \alpha=\frac{-1+\sqrt{5}}{2} and \beta=\frac{-1-\sqrt{5}}{2}, i.e. the roots of x^{2}+x-1 Check that f(x)=\frac{1}{\sqrt{5}(x-\alpha)}-\frac{1}{\sqrt{5}(x-\beta)}...
5. ### Recursive sequence limit challenge

Hi, I saw this problem in Coursera's Calculus II by the Ohio State University: Most of us are familiar with the Fibonacci Sequence With\quad F_{ 0 }={ F }_{ 1 }=1\\ \\ { F }_{ n }={ F }_{ n-1 }+{ F }_{ n-2 }\\ \\ \left\{ 1,1,2,3,5,8,13,21,...{ F }_{ n } \right\} Now we can build { G }_{ n...
6. ### Derive Binet's Formula

Hello! I need to derive Binet's Formula! I know how to do it for one of the definitions of the Fibonacci sequence: F(1)=1, F(2)=1, F(n)= F(n-1)+ F(n-2), for all n > or = 3. However, for my assignment, I have to use an alternate definition: F(0)=1, F(1)=1, F(n)= F(n-1)+ F(n-2) for all n > or...
7. ### Fibonacci induction

Hello I need some help with this problem For each natural number n, let fn be the nth Fibonacci number. Prove that (f_(n+1))^2 + (f_n)^2 = f_(2n+1). so far what I have let n=1 so 1^2 + 1^2 = 2 which is true so the equation holds for n= 1 (base case) then I pick an arbitrary k where...
8. ### Limit of ratio of Non-consecutive terms of Fibonacci Sequence

Hi folks, If anyone can help with this 'd be delighted. I can prove that the limit of the ratio of consecutive terms of the FS is PHI. However, how do I go about proving that the limit of NON-consecutive terms is phi to the power of t eg Lim n--->inf (Un/Un+t)=Phi^t Any help would be...
9. ### Fibonacci Sequence Formula

Finding Formula for given sequence I have the following sequence: 0 2 3 5 8 13 21 34.......... This is very close to the fibonacci sequence, but not exactly. Any ideas?
10. ### Fibonacci generating functions for pair/impair indexes.

Let f_n be the sequence of Fibonacci numbers. It's known that \sum_{n \ge 0} f_n x^n = \dfrac{1}{1-x-x^2} What is the interpretation of this for \sum_{n \ge 0} f_{2n} x^n \text{ and }\sum_{n \ge 0} f_{2n+1} x^n\text{ ?} How do we prove it?
11. ### Help with proof regarding fibonacci sequence

The Fibonacci sequence is given: A1 = A2 = 1 , An = An-1 + An-2 For m >= 1 and n >= 1 Prove that Amn is divisible by Am. I have already prepared and proved by induction a Lemma which is: For m >= 2 and n > = 1. Fibonacci sequence satisfies: Am+n = Am-1An + AmAn+1 The consequence of the Lemma...
12. ### Euclid's algorithm and fibonacci numbers

Let F_{0}, F_{1}, F_{2}, F_{3},... be the Fibonacci numbers, defined by F_{0}=1 F_{1}=1 F_{n}=F_{n-1}+F_{n-2} for n\geq 0 a) Prove that for all n\geq 0 we have gcd(F_{n+1}, F_{n})=1 b) Prove that for all n\geq 0 we also have gcd(F_{n+2}, F_{n})=1 (Check what Euclid's algorithm would do if...
13. ### Fibonacci Proof by Induction

Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n. I have gotten to the inductive goal when plugging in k+1, but am lost from here. Thanks.
14. ### Fibonacci problem, proof by induction?

Given: alpha = (1+ sqrt5)/2 and beta = (1-sqrt5)/2 alpha^2 = 1 + alpha and beta^2 = 1+ beta Use induction to prove that for all integers n >= 1 we have alpha^n = f(n-1)+ f(n)*(alpha) and beta^n = f(n-1)+ f(n)*(beta) I've did my base case and plug in k+1 to n but I can't get them equal to each...
15. ### Fibonacci Proof Help

Consider the recursion Fk = Fk−1 + Fk−2, k ≥ 2, with F0 = 0 and F1 = 1. Prove that Fk is even if and only if 3 | k. In other words, prove that, modulo 2, F3t = 0, F3t+1 = 1, and F3t+2 = 1 for t ≥ 0. I'm just having some trouble solving the above proof. I see why it works (because odd + odd =...
16. ### Fibonacci sequence

Hi can anyone explain how i can get express Fn+3 in terms of Fn+2 and Fn to show that Fn+2 = 1/2(Fn+3 + Fn). Any help is much appreciated
17. ### Fibonacci Sequence question

Hey all, Im hoping you guys could point me in the right direction after getting myself rather lost with a question on my assignment. I am not putting the actual question as i would like to work it out myself. here is a question i have made up that is similar with what i think to be correct...
18. ### Fibonacci Functions and Sets (+Java)

I would like some help getting through these 2 problems: Implement all three methods we discussed for computing a given element F_n the Fibonacci sequence: F_0 = F_1 = 1. F_k = F_k-1 + F_k-2 for k > 1. 1. Recursive (write a recursive function to compute F_n given input n) 2. Inductive...
19. ### Proof the following proposition: for all n ≥ 0 , fib(n) ≤ n!

Hi All, I am a comp science undergrad and just started to learn proof. And I have been thinking about this question for a few days. How should I present my answer? Do I have to use the Binet's formula? Or can I do this: Base case: let n = 0 fib(0) = 0 0! = 0 Thus, fib(n) ≤ n! is true when...
20. ### Fibonacci divisibility proof (factors of fibonacci numbers)

I investigated the factors of fibonacci numbers and found that one factor always was a fibonacci number, for example the fibonacci number of 2 has 2 factors 1x2, and the factor 2 is a fibonacci number. Now I have to make a proof for this: I think this site explains it but I dont fully understand...