# Tagged Questions

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 $F_1=1$.

24 views

### Closed form expression for zero of recurrence relation

Given the recurrence $d(i+1)=xFib(2i+1)-nFib(2i)$, where $Fib$ denotes the Fibonacci sequence (i.e. $Fib(0)=0, Fib(1)=1, Fib(2)=1, Fib(3)=2$, etc) and $n$ and $x$ are arbitrary integers, is it ...
46 views

### How to prove this sequence is null?

I am working on the fibonacci numbers series using the ratio. To prove convergence I want to show that the sequence of the series is going to 0. And then according to the Leibniz criterion the series ...
37 views

### Fibonacci numbers and binomial theroem [duplicate]

So I am trying to prove $$\sum_{i=0}^n{nCi×F_i} = F_{2n}$$ Such that $$nCi = \frac {n!}{i!×(n-i)!}$$ And $F_i$ is the ith value of the fibonacci sequence such that $F_0 = 1$ and $F_1 = 1$ I have ...
60 views

### Proof: Fibonacci Sequence (2 parts)

Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers. I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely ...
31 views

### Divisibility of Fibonacci Sequence mod prime

I have to solve the following problem and I have a few questions: Consider the Fibonacci sequence defined as $F_n:=2F_{n-1}+F_{n-2}$ with $F_0=1$ and $F_1=1$. Now, I need to prove that for any odd ...
119 views

37 views

### Sum of Squares for Odd Fibonacci Numbers

I am trying to prove the following theorem by induction: THEOREM: For the Fibonacci sequence $F_1$, $F_2$, ... , $F_n$ defined as, $F_1$ = $F_2$ = 1 $F_n$ = $F_{n-1}$ + $F_{n-2}$ for n >= 3, For ...
125 views

### Proving that every integer has a Fibonacci number multiple

Show that for any positive integer, there exists a Fibonacci number N such that N is divisible by the integer. I'm not really sure how to begin my approach to this problem, would really appreciate ...
31 views

58 views

### The sum of the Reciprocal of the Partial Sum of the Consecutive Fibonacci Numbers Series [closed]

How to prove that this conjecture is true for $n$th order Fibonacci number: $$1\le\sum_{n=1}^\infty\dfrac{1}{F_{n+2}-1}<2.5$$
58 views

### Calculating number of tile sequences

My daughter (aged 12) came to me with the problem below. I was able to help her to some extent but I could not see an age-appropriate solution. That is, I could imagine solutions involving factorials /...
### Are Fibonacci numbers with a square prime index always divisible by $F_p$?
I am doing some research on sequences and I need some help. The sequence of $F_{p^2}$ seems sort of different. It seems that because the index only has one distinct prime factor, as a result the only ...