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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Sum of inverse of Fibonacci numbers

If $F(n)$ is the nth Fibonacci number, How can I prove that: $$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$
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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: $$1\le\sum_{n=1}^\infty\dfrac{1}{F_{n+2}-1}<2.5$$
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Use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ and write it as a power series

Find the roots $α_1$, $α_2$ of $x^2 + x – 1$ and use partial fractions to write $\frac{1}{x^2 + x – 1}$ as $\frac{A_1}{x – α_1} + \frac{A_2}{x – α_2}$ , for suitable $A_1, A_2$. Using the power series ...
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Prove that $\frac{a_{n+1}}{a_n}<2$ for every n>1 using induction

Fibonacci sequence of $a_n$: Prove that $\frac{a_{n+1}}{a_n}\leq2$ for every $n\geq1$. I was able to prove this using the base case: $$n=1 | n=2$$ $$\frac{a_{n+1}}{a_n}\leq2|\frac{a_{n+1}}{a_n}\leq2$$...
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Prove that for all $n \geq 1$, $F_{-n}$ = $(-1)^{n+1}F_n$ where F is the Fibonacci numbers.

Prove that for all $n \geq 1$, $F_{-n}$ = $(-1)^{n+1}F_n$ where F is the Fibonacci numbers. I've already shown that the formula holds for $n = 1$ and $n = 2$. So I supposed the formula holds for $n$ ...
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Proof Fibonacci derivation

I was wondering how to prove that $$f(n+m+2) = f(n+1)f(m+1) + f(n)f(m)$$ where $f$ is the fibonacci sequence and n, m are positive integers. Can be this done with induction? I'm lost with this ...
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Variations on Fibonacci Sequence

Do mathematicians use variations on the Fibonacci sequence? I'm thinking specifically about something like this: Start with three $1s$ and for each consecutive number, add the three previous number ...
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Is $\sum_{n,m \geq 0} F_n^m x^n y^m$ a rational generating function?

I am curious if the generating function defined by: $$F(x,y)=\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} F_{n}^m x^n y^m$$ where $F_n$ is the $n$th fibonacci number, is a rational function. That is, Is ...
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Proof that Fibonacci Sequence modulo m is periodic? [duplicate]

It's well known that the Fibonacci sequence $\pmod m$ (where $m \in \mathbb N$) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more ...
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Sum of digits of Fibonacci number a perfect square

During my problem solving with Fibonacci numbers following thought crossed my mind. How many Fibonacci numbers are there such that sum of its digits is a perfect square? Here is a list of ...
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Basic Discrete Mathematics Recurrence question

Good afternoon, I've been assigned the following problem from my Intro to Discrete Mathematics: Show that $\sum_{i=1}^n$ F(i) = F(n+2) - 1 note: F(n) is the nth term in the fibonacci sequence. I'...
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Find a generating function with Fibonacci

$$G(x) = \sum_{n=1}^\infty na_n x^n$$ Hello. I need to find a generating function for the summation above, where $a_n$ is the Fibonacci sequence. I have found the generating function for the fib ...
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Greatest Common Divisor with Fibonacci Numbers [duplicate]

Prove that for all integers $n\geq 0$: $$\gcd(F_{n+1},F_n)=1$$ I am extremely lost. Please can some provide some hint or direction? Thank you so very much
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Applying Fibonacci Fast Doubling Identities

So I sort of understand of how these identities came about from reading this article. $F_{2n+1} = F_{n}^2 + F_{n+1}^2$ $F_{2n} = 2F_{n+1}F_{n}-F_{n}^2$ But I don't understand how to apply them....
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Fibonacci Numbers Induction?

Show that $a_n=n^2+n+1$ satisfies \begin{cases} a_0=1\\ a_k=a_{k-1}+2k & \text{for $k>0$} \end{cases} I want to use induction to solve this problem. but I don't know what my base will be ...
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Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$

As stated in the tag, I'm trying to prove by induction the claim $F_{2n} = F^2_{n+1} - F^2_{n-1}$, where $F_{n}$ is the $n^{th}$ Fibonacci number. I've spent hours on the inductive step without ...
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Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
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Rate of Convergence vs Radius of Convergence

What is the difference between finding the 'rate of convergence' and the radius of convergence'? The question I am trying to solve here is to find the rate of convergence of the ratio of Fibonacci ...
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Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$

Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$; I was stuck with this question for a while... Help me please!!! Thanks!!!
On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers
While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
I found on Wikipedia the following infinite sum : $$\sum_{k=0}^{\infty} \frac{1}{1+F_{2k+1}}=\frac{\sqrt{5}}{2}$$ There is no reference for this sum in the article and I couldn't find it ...