Linked Questions

30
votes
3answers
2k views

limit $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$

$\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$ I tried to write it as $\sin (n\pi - \theta)$ to get the form $∞-∞$ form within $\sin$ function. But could not proceed after that. How should I do it? Edit:I am ...
25
votes
7answers
20k views

How to prove that the Binet formula gives the terms of the Fibonacci Sequence?

This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$. Show that ...
28
votes
3answers
7k views

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational. I can prove that $x$ is irrational by showing that it's a root of a polynomial with integer coefficients ...
7
votes
6answers
6k views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and I'...
4
votes
6answers
448 views

Show that $\forall n\in\Bbb{N}, (3+\sqrt 7)^n+(3-\sqrt 7)^n\in\Bbb{Z}$ and that $\forall n\in\Bbb{N}, (2+\sqrt 2)^n+(2-\sqrt 2)^n\in\Bbb{Z}$

I got this problem which I encountered during a limit of sequence calculation: Show that $\forall n\in\Bbb{N}, (3+\sqrt 7)^n+(3-\sqrt 7)^n\in\Bbb{Z}$ And that $\forall n\in\Bbb{N}, (2+\sqrt 2)^n+(2-\...
5
votes
3answers
344 views

Prove that $\left(\frac{3+\sqrt{17}}{2}\right)^n + \left(\frac{3-\sqrt{17}}{2}\right)^n$ is always odd for any natural $n$.

Prove that $$\left(\frac{3+\sqrt{17}}{2}\right)^n + \left(\frac{3-\sqrt{17}}{2}\right)^n$$ is always odd for any natural $n$. I attempted to write the binomial expansion and sum it so the root ...
11
votes
2answers
253 views

Why is $\frac{1}{\sqrt 5}\left[\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right]$ an integer? [duplicate]

I am looking for a proof on why $$\frac{1}{\sqrt 5}\left[\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right]$$ an integer. I have seen many proofs on this, but they all ...
8
votes
1answer
1k views

How to prove that Fibonacci number is integer? [duplicate]

How to prove that formula for Fibonacci numbers are always integers, for all $n$: $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{...
3
votes
4answers
217 views

Hint to prove that $\phi^n + \phi'^n$ is an integer.

I was solving some induction exercises but I found this that I could not solve. Let $n \in \mathbb{N}$, prove that $\phi^n + \phi'^n$ is an integer where $\phi=\frac{1+\sqrt{5}}{2}$ and $\phi'=\...
0
votes
2answers
3k views

General Solution to $x^2-2y^2=1$ [duplicate]

Find a general solution to $x^2-2y^2=1$ I found that (3,2) is a solution. Now what should I do? I can not catch what the question really want. It is about pell's equation. Would you give me a form ...
3
votes
2answers
182 views

Fibonacci General Formula - Is it obvious that the general term is an integer? [duplicate]

Given the recurrence relation for the Fibonacci numbers, $F_{n+1}=F_{n}+F_{n-1}$ with $F_0=1$ and $F_1=1$ it's obvious that $F_n$ is a positive integer for all $n$. Suppose instead we were given $$F_n=...
4
votes
2answers
107 views

How to write $[(2+\sqrt{3})^n + (2-\sqrt{3})^n + (2+\sqrt{3})^{n+1} + (2-\sqrt{3})^{n+1}]/6$ to the form $a^2 + 2 b^2$ ($a, b \in \mathbb{N}$).

We know that $(2+\sqrt{3})^n + (2-\sqrt{3})^n$ is an integer (See here). However, we want to write the formula \begin{align} &\frac{3+\sqrt{3}}{6} (2+\sqrt{3})^n + \frac{3-\sqrt{3}}{6} (2-\sqrt{3}...