# Tagged Questions

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### How to solve this recurrence relation consisting of 9 equations (one of them with a minimum function)?

Given the recurrence relation as follows. $T_{k}=min((W_{k}+Y_{k}) , (X_{k}+Z_{k}))$ $A_{k}=T_{k}\frac{W_{k}}{W_{k}+Y_{k}}\frac{X_{k}}{X_{k}+Z_{k}}$ ...
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### Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
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### Dynamics of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and maybe easy question, but I was not able to find an answer. If $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a linear change of coordinates it can be brought to the ...
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### How to solve this recurrence of a sequence?

$x_1=1$, $x_2=2$, $x_n=n\cdot x_{n-1}+\frac{n(n-1)}{2}x_{n-2}$ Thanks to Barry Cipra's suggestion, now I have obtained the solution for $x_n$: ...
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### if $(n+1)c_{n}=nc_{n-1}+2nd_{n-1},d_{n}=2c_{n-1}+d_{n-1}$How prove $c_{n},d_{n}$ are integers

let sequences $c_{n},d_{n}$ such $$c_{0}=0,d_{0}=1$$ and for $n\ge 1$,have $$c_{n}=\dfrac{n}{n+1}c_{n-1}+\dfrac{2n}{n+1}d_{n-1}$$ $$d_{n}=2c_{n-1}+d_{n-1}$$ show that $c_{n},d_{n}$ are integers. my ...
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### Summation of logarithmic series

I am solving a recurrence relation and it requires me to sum the following series upto $\log{n}$ terms - $1/\log(n) + 1/\log(n/2) + 1/\log(n/4)$..... The base in each term is $2$. Any help on ...
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### Prove any $k\in\mathbb{N}$ can be created using sum of $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$

I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition. My attempt was just bad... I've ...
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### Recursive sequence with binomial coefficients

I have a sequence $\epsilon_i$ defined recursively for $i\ge 1$ as follows \begin{eqnarray*} \epsilon_1 &=& \frac{1}{p}\\ \epsilon_n &=& \frac{1}{1-(1-p)^n}\left( 1 + \sum_{j=1}^{n-1} ...
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### Does there exist an infinite sequence $p_0,p_1,p_2…$ of prime numbers such that $p_k=4 p_{k-1}\pm 1$

$k \in Z^+$ firstly we know that there exists infinetly many primes of the form $4n+1$ by FTA also we see that if we consider finite primes say to $n$ then the recursive formular can be expressed ...
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### Solving recurrences with summation factors (Concrete Mathematics)

Chapter 2 in Concrete Mathematics talks about solving recurrences of the form $$a_{n}T_{n}=b_{n}T_{n-1}+c_{n}$$ by reducing them into a sum. The authors multiply both sides by a summation factor ...
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### Recurrence equation for $(-1)^k k$

In a project of mine I came across the recurrence relation $$a_{n+1} = 1 -(n+1)\sum_{k=1}^n{\frac{a_k}{n-k+1}\binom{n}{k}},\quad a_1=2;$$ From calculating the first few terms it seems obvious that ...
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### representing a recursive difference equation of two variables into one variable equation

suppose the following recursive difference equation ($t$ is time): $$x_t = \frac{a}{1+a}x_{t-1} + \frac{1}{1+a}x_{t+1}$$ where $0<a<1$ is assumed and all values of $a$ at past times are ...
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I'm posting this question because this is new material for me and I am unsure of my answers and have no one to consult with. I solved the first three and would appreciate feedback. I need help solving ...
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### Repertoire Method. How to know when equations are valid

I'm trying to work a few examples from the Repertoire Method from the Book Concrete Mathematics. I'm working through the following recurrence: $f(0) = 1$ $f(n) = 2f(n-1) + n$ Which I generalized ...
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### Equivalent of a recurrence sequence [duplicate]

Let $x_{0} = 2$ and $x_{n+1} = x_{n} + \ln(x_{n})$, how can I find an asymptotic equivalent of this sequence say, to the third term? (This is not homework, it was a problem in the Oral Examination ...
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### Convergence of a sequence with repeated sines [duplicate]

Let $x\in(0,\pi/2)$ and $\{a_n\}_{n\in\mathbb N}$ defined recursively as follows: $$a_0=x, \quad \text{and} \quad a_{n+1}=\sin(a_n).$$ Show that $$\lim_{n\to\infty}{n\,a_n^2}=3.$$ Note. There is ...
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### Constructing $N$ unit cubes

I was trying to solve the problem of construction $N$ unit cubes, and while searching I came across this sequence at OEIS. This is exactly what I need but I could not find a method to generate the ...
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### Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
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### Limit of a fraction of double factorials

How can we show that \begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*} where \begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*} ...
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### A self-convolution formula that counts bracket expressions

Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \$ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this ...
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### Concrete Mathematics - Stability of definitions in the repertoire method

There are some existing questions on the repertoire method from the first chapter but I think I'm stuck on something different than the part people usually have trouble with. I think the jump in the ...
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### Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as a(n) = a(n-1) - 1/a(n-1), where a(0) is given. ...you already know what I'm asking you ...
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### Limits of a recursively defined sequence [closed]

Let $x_1=a$ and define a sequence $\left(x_n\right)$ recursively by: $x_{n+1} = \dfrac{x_n}{1 + \frac{x_n}{2}}$ For what values of $a$ is it true that $x_n$ approaches $0$?
Can you please help, my son has been trying for over two hours now to solve the following: A sequence of terms $\left\{u_n\right\}$ is defined for $n\geq 1$, by the recurrence relation: ...