# 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 $F_1=1$.

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### Last 10 digits of the billionth fibonacci number?

I want to compute the last ten digits of the billionth fibonacci number, but my notebook doesn't even have the power to calculate such big numbers, so I though of a very simple trick: The carry of ...
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### Fibonacci series generated with division $1/999999999999999999999998999999999999999999999999$ [duplicate]

I tested the suggestion of finding the Fibonacci series by division, which sounded very surprising to me. I therefore used a simple sympy script to test it and found that it works as advertised. ...
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### Proving a Recursive Formula

I know there are some questions on this site about how to find a recursive formula, but I've already found the formula. I'm doing an assignment (http://mathstat.dal.ca/~svenjah/math2112/Assign6.pdf) ...
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### Covering a rectangle of size $n\times1$ with dominos

A rectangle of size $n\times1$ is given. (a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$? (b) In how many ways the rectangle can be covered with ...
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### Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
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### Sum of odd Fibonacci Numbers

Trying to prove that the sum of odd-index consecutive Fibonacci numbers is the next even-index Fibonacci number. I have a gap in my proof that I cannot figure out. I know that induction would be ...
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### Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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### Discrete Mathematics Fibonacci Sequence

I am studying for the final exam in my Discrete Mathematics class and came upon the following problem on the study guide we were given. Given the following algorithm: If $n = 0$, then $f(n) = 0$ ...
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### Why do I get the Fibonacci sequence when I start with 1 and keep using the + sign?

I played with this on a calculator and when I entered 1 and kept hitting the + button, I got the noticeable Fibonacci sequence! Can someone explain to me why this happens?
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### Proving that for Fibonacci numbers $a_n \lt (\frac {1 + \sqrt 5} 2)^n$ for $n \ge 1$

I'd like to prove that for Fibonacci numbers $a_n \lt \left(\frac {1 + \sqrt 5} 2\right)^n$ for $n \ge 1$. I suppose it needs induction so, after verifying the trivial case $n=1$, the inductive step ...
I've been working on this following problem: Find a constant $c< 1$ such that $F_n \leq 2^{cn}$ for all $n \geq 0$. I honestly have no idea where to begin on this. I've done plenty of proofs ...
I have to prove that for any natural number $n$ there exists $i>0$ such that $n\mid F_i$, where $F_i$ is the $i$-th Fibonacci number.