Questions tagged [fibonacci-numbers]

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 $F_1=1$.

The $n$th Fibonacci number $F_n$ is defined recursively, by

$$F_n = F_{n - 1} + F_{n - 2}$$

for $n > 1$, and $F_0 = 0,\; F_1 = 1$. There is a closed form expression, namely

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$$

where the golden ratio $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$.

Combinatorial identities involving the Fibonacci numbers have been extensively studied, and the numbers arise frequently in nature and in popular culture.

Reference: Fibonacci number.

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Fibonacci sequences and related series

Let $\{a_n\}$ be a sequence such that $a_1=a_2=1$ and $a_{n+1}=a_n+a_{n-1}$ for $n\geq 2$. Prove that $\displaystyle \sum_{n=1}^\infty \frac{1}{a_n}$ converges. My work: Let $b_n=\frac{1}{a_n}$. Then I proved that $|\frac{b_{n+1}}{b_n}|<1$. I was…
Extremal
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Divisibility of $987x^n − F_nx^{16} + F_{n−16}$

If $F_n$ is $n^{th}$ Fibonacci number, and polynomials $P_n(x)$ are defined as $987x^n − F_nx^{16} + F_{n−16}$, prove that for all $n ≥ 1$, $P_n(x)$ is divisible by $x^2−x−1$. This is from a contest prep class. The hint was to use induction, but…
VividD
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Fibonacci, prove that $F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$ with induction

I need to prove by induction that: $$F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$$ I did the following: Check if the statement holds for $n=1$: $$1\cdot 3-(2)^2=(-1)^1$$ Check if the statement holds for $n=p+1$: $$F_{p+1}\cdot…
Adnan
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Confusion on unberstanding the proof of induction regarding Fibonacci numbers

I am trying to understand the proof that "For all $n\geq 2, F_n^2-F_{n+1}F_{n-1}=(-1)^{n-1}$.Where $F_n$ stands for the Fibonacci number at $n$. I got this proof from a book and here is the proof. Fn+12 = Fn+1 (Fn+Fn-1) by the definition of…
User1
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Matrices, determinants, and applications to identities involving Fibonacci numbers

Preamble It is well known that since: $$ \begin{pmatrix} F_{n+1} \\ F_n \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix} \begin{pmatrix} F_n & F_{n-1} \\ \end{pmatrix} $$ it is valid that: $$ \begin{pmatrix} 1 &…
VividD
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Summation of a multiple series involving Fibonacci numbers

Compute the sum $$\sum_{a_{2015} = 0}^{\infty} \sum_{a_{2014} = 0}^{a_{2015}} \sum_{a_{2013} = 0}^{a_{2014}} \cdots \sum_{a_{1} = 0}^{a_2} \sum_{k=0}^{a_1} \frac{F_{k}}{2^{a_{2015}}} $$ where $F_k$ denotes the $k$th Fibonacci number ($F_0 = 1$,…
user203482
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Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number

If $F_n$ is the $n$-th fibonacci number, then prove that, $$F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$$ I tried the idea of using Pascal's triangle, but it seems to need some adjustment, so I am stuck, please help. Any…
Swadhin
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A series for Fibonacci numbers.

How can I prove The Fibonacci sequence is encoded in the number $1/89$ i.e. $( 1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034 \ldots)$
Bhaskar Vashishth
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Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = \frac{\varphi^z - \cos(z…
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How to prove this equation by induction?

I am trying to prove this equation by mathematical induction $$f_{n+1}f_{n-1} = f_{n}^{2}+(-1)^n$$ is true where $f_{n} = $ the nth number in the Fibonacci sequence. I don't quite get how to do this for the case of $n+1$. Can anyone provide a…
SalmonKiller
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Prove that the set of solutions to $F_{n+2} = F_{n+1} + F_n$ is of dimension 2

I was playing with the Fibonacci sequence, willing to prove that $$ F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right) $$ I did the usual, looking for geometric sequences satisfying the…
Jerome
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Discrete Math - Given n>= 5, how many times does fib(4) occur?

I have been trying to solve the problem below (and similar problems) but I have no clue how to tackle it. Can you please help me tackle this particular problem, and how to attack similar problems? The question in question is question 11 at this…
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Prove that the Fibonacci recursion diverges

I have this sequence with $ n \in \mathbb{N} $ $ f(1) = f(2) = 1 $ and $ f(n) = f(n-1) + f(n-2) $ for $n \ge 3$ I think this sequence is bounded below and unbounded above. So it's clear that this recursive sequence diverges. Questions: Is this…
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Prove by induction fibonacci variation

Prove by induction: The fibonacci sequence is defined as follows: $f_1 = 1$, $f_2 = 1$ and $f_{n+2} = f_n + f_{n+1}$ for $n \geq 1$ Prove by induction that $f_1^2 + f_2^2 + \dotsb + f_n^2 = f_n f_{n+1}$ I am new to proofs by induction and having a…
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Induction proof for Fibonacci numbers

I am trying to get a hang on the induction method for proof, but I'm still dubious of many aspect of this proof regarding its application to sequences of integers, such as the Fibonacci sequence. QUESTION: Prove by induction that this is true.…
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