$$ z = 1 + 2i \ (complex \ number) \\ z^n = a_n + b_ni \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) \\ We \ know \ that: \ b_{n+2} - 2b_{n+1} + 5b_n = 0 \\ a_{n+1}=a_n-2b_n \\ b_{n+1}=b_n+2a_n $$  Prove that $ z^n \notin \mathbb{R} $  What I tried:  We know that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$  If $ z^n \in \mathbb{R} $ it means that $ b_n = 0 $  So,  $ b_{n+2} - 2b_{n+1} = 0 $ and I tried to get a contradiction but I can't because I get $ -3b_n = 0 $.  How can I solve this problem? Thank you!
$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$  How can I solve this? Thank you!  EDIT: Or please tell me your ideas.
Is there a way to draw a sort of connection between subplots in TiKz/pgfplot, like in this ? I'm looking to achieve something like this:    But I would need to draw the "x 2000" line between the plots.  EDIT: So right now I have something like this:    Coming from my main.tex:  \input{aipcheck}  \documentclass[final]{aipproc}  \layoutstyle{8x11single} \usepackage[utf8]{inputenc} \usepackage{algorithm2e} \usepackage{color} \usepackage[usenames,dvipsnames]{xcolor} \usepackage{cleveref} \usepackage{pgfplots}  \pgfplotsset{ tick label style={font=\tiny}, %axis numbers label style={font=\small}, %axis labels legend style={font=\tiny} }  \begin{document}  (...)  \begin{figure}     \centering     \newlength\figureheight      \newlength\figurewidth      \setlength\figureheight{5cm}      \setlength\figurewidth{5cm}     \input{figs/_ddmp2d.tikz} \label{fig2D} \end{figure}   with _ddmp2d.tikz, originally created from matlab2tikz, but already modified, containing:  % This file was created by matlab2tikz v0.3.3. % Copyright (c) 2008--2013, Nico Schlömer &lt;nico.schloemer@gmail.com&gt; % All rights reserved. %  % The latest updates can be retrieved from %   http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz % where you can also make suggestions and rate matlab2tikz. %  %  %   \begin{tikzpicture}  \begin{axis}[% width=\figurewidth, height=\figureheight, scale only axis, xmin=0, xmax=15, xlabel={Number of iterations}, ymin=0, ymax=16000000000000, name=plot1, after end axis/.append code={\coordinate (a) at (axis description cs:1,0);}]  \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 25160000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 52501000000\\ }; \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 5324200000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 551920000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 3 0\\ 3 16000000000000\\ };  \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 4 0\\ 4 16000000000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 5 0\\ 5 16000000000000\\ }; \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 8 0\\ 8 16000000000000\\ }; \end{axis}  \begin{scope}[xshift=5cm] \begin{axis}[% width=0.35\figurewidth, height=\figureheight, scale only axis, xmin=995, xmax=1000, xtick={997,998,999,1000}, xticklabels={$997$,$998$,$999$,$1000$}, ymin=0, ymax=8000000000, at=(plot1.right of south east), anchor=left of south west, yticklabel pos=right, axis x discontinuity=crunch, after end axis/.append code={\coordinate (b) at (axis description cs:0,1);} ] \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt ] table[row sep=crcr]{ 995 6093100000\\ 996 6093100000\\ 997 6093100000\\ 998 6093100000\\ 999 6093100000\\ 1000 6093100000\\ };  \addplot [ thick, color=black, dashed ] table[row sep=crcr]{ 995 1577100000\\ 996 1598000000\\ 997 1598500000\\ 998 1628600000\\ 999 1631000000\\ 1000 1595900000\\ };  \addplot [ thick, color=blue, dotted ] table[row sep=crcr]{ 995 2592000000\\ 996 2591900000\\ 997 2591700000\\ 998 2591600000\\ 999 2591500000\\ 1000 2591400000\\ };  \addplot [ thick, color=green, solid ] table[row sep=crcr]{ 995 391810000\\ 996 391810000\\ 997 391810000\\ 998 391810000\\ 999 391810000\\ 1000 391810000\\ };  \end{axis} \draw[gray] (a) -- (b) node[pos=0.5,sloped,fill=white] {$\times$2000}; \end{scope}  \end{tikzpicture}% Query Activity for ORDER BY xxx DESC LIMIT To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$ What is the reason behind the multiple Spider-Man reboots? I have realized that to pronounce the plural form of words ending in -th, we have to drop the letter "h". e.g. Months is pronounced /mʌnts/. But I have never seen this point mentioned in any context. I just want to make sure that this rule is correct and can be always applied. Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true? I want to prove that $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ is convergent.  Therefore I want to use that it will be convergent if $\lim\limits_{n \to \infty}|\frac {2^{n+1}} {n^n}| &lt; 1$ is true.  But I am not sure if all of my steps are legal to do in order to do it. Here is what I came up with:  $$\lim_{n \to \infty}|\frac {2^{n+1}} {n^n}| &lt; 1$$ $$\lim_{n \to \infty}|\frac {\sqrt[n]{2^{n}} \sqrt[n]{2}} {\sqrt[n]{n^n}}| &lt; \sqrt[n]1$$ $$\lim_{n \to \infty} \frac {2 \sqrt[n]2} {n} &lt; 1$$ $$0&lt;1$$  $\Rightarrow$ It is true. I installed Ubuntu 12.10 and entered a password in SeaHorse. I would like to remove the password of "Default" but I get this error :  GDBus.Error:org.freedesktop.Secret.Error.NoSuchObject: The collection does not exist   Is there a fix or a way to bypass modify the password ? I am trying to test the impact of a sale initiative on store revenue. I have created 10 matched pairs of stores and then randomly distributed one store in each pair into a test or control group.  I then ran the sale for 21 days within the test group (keeping control constant) and measured relative lift in revenue compared to baseline.  My questions  1.) Should I be using a paired t-test based on the previous store matching  2.) Should I be measuring across each day of the test such as:          Control Test 1/1/2017    4   3 1/2/2017    3   4 1/3/2017    1   3 1/4/2017    2   3 1/5/2017    2   4 1/6/2017    1   7   Or measuring across my paired markets such as:                 Test Control Market Pair 1   10  5 Market Pair 2   8   3 Market Pair 3   5   1 What IDE to use for Python? Short story - "One man's meat is another man's poison" Pros and cons of rooting using apps ("Soft Root") compared to other methods ("Hard Root") There are quite a few app based rooting methods. A recent review , points to some of them and there may be more apps, paid or otherwise.  From what I understand,   Plus Points   Ease of rooting Don't need a laptop or computer   Minuses   Based on exploits, so may not work if exploits are denied by OS updates Difficulty in unrooting (as I see on some forums for my device Huawei Honor 6)   Questions:    What are the pros and cons apart from above?  If a device has both options - app based rooting and rooting by methods by developers, which one should I opt for?   Note: I am not seeking app suggestion or recommendation. Sci-fi story where all 'citizens' had shares in a global company Not everyone was a citizen though, there was a significant underclass who had no shares.  One character was a celebrity in some sort of combat game who was followed by camera drones &amp; used drugs to suppress puberty so he waa an eternal 'child', I have the impression that medical science allowed virtual immortality short of accidents.  One character (may have been the same one) had had a 'previous life' &amp; had committed personality suicide by some form of mind wipe bequeathing his worldly possessions to the new personality that would eventually develop in his old body.  I think the motive for this 'suicide' was essentially boredom, he'd lived a long time.  The mind wipe in question was pretty extensive &amp; all encompassing, he had to be cared for &amp; educated from scratch as if a newborn baby in an adult body, nappies &amp; all (but I'm pretty sure this was purely backstory in the book I read).  He also had miniature nano organisms (or bots?) like miniature worms or millipedes that complimented his immune system &amp; aided healing, they were self replicating, pretty sure they were some sort of bioengineered organism tailored to his DNA rather than bots  but not 100% on that.  There was an organism spanning a large area on the planet that absorbed (or consumed) anything that wandered into it but saved the DNA of anything it had previously absorbed &amp; might randomly reproduce any animal plant or individual it had previously absorbed fully formed.  One of the characters fell into this but got pulled out before the full effects took hold, as a result he grew leaves from his head &amp; his immune system was then constantly battling with the 'infection'.  When did I read this? probably some time in the 80's in the UK.  When was it written? I haven't a clue, sorry.  It was a paperback, probably something I had out of a library, because I never get rid of anything I've bought myself so I'd still have it &amp; be able to dig it out if it wasn't a library loan.  I have lots of fragments of world background &amp; 'colour' that seem to have stuck in my mind there, but a complete lack of any story arc to go with them. Show that there's no continuous function that takes each of its values $f(x)$ exactly twice. I need to prove the following:  There's no continuous function $f:[a,b]\to \mathbb{R}$ that takes each of its values $f(x)$, $x\in [a,b]$  exactly twice.  First of all, I didn't understand the question. For example $x^2$ takes $1$ twice, in the interval $[-1,1]$. Is it saying that it does not occur for all $x$ in the interval? But what about $f(x) = c$? Is it saying that it does not occur only exactly $2$ times, then? I have no idea about how to prove it. I know that for $f(x)$ such that $f(a)&lt;f(x)&lt;f(b)$, if $f$ is continuous then there is a $c\in [a,b]$ such that $f(c) = f(x)$.  Now, there's the following proof in my book and I really wanted to understand it, instead of just getting a new proof  Since the interval $[a,b]$ has only $2$ extreme points, then the maximum or minimum of $f$ must be in a point $c\in int([a,b])$ and and in another point $d\in [a,b]$. Then, there exists $\delta&gt;0$ such that in the intervals $[c-\delta, c), (c,c+\delta)$ (and if $d$ is not extreme of $[a,b]$, $[d-\delta, d]$) the function takes values that are less than $f(c) = f(d)$. Let $A$ be the greatest of the numbers $f(c-\delta), f(c+\delta), f(d-\delta)$. By the intermediate value theorem, there are $x\in [c-\delta, c), y\in (c, c+\delta]$ and $z\in [d-\delta, d)$ such that $f(x)=f(y)=f(z)=A$. Contradiction.   Well, why the last part? Why is it that I can apply the intermediate value theorem to these values? For example, $&lt;f(c-\delta)&lt;p&lt;f(c)$, then by the theorem I know that there exists $m\in [c-\delta, c)$ such that $f(m) = p$. Same for the other intervals. But what guarantees thhat the greatest of the values between  $x\in [c-\delta, c), y\in (c, c+\delta]$ will be inside the intervals $[c-\delta), c), (c,c+\delta), [d-\delta, d)$? Iterate until condition is met Curved space-time VS change of coordinates in Minkowski space How to access blocked sites? I am in Pakistan and  is blocked here. I want to take the Lecture videos from YouTube. In Windows, I was using  to bypass the proxy.   But now, using Ubuntu, I don't know how to bypass the proxy. I found some where that i can bypass the proxy by using putty software. Can you guide me how can I bypass proxy by that? I tried, but I was not able to do so. Kindly tell me some easy method to bypass proxy. I don't want to used websites like accesstoblockedsites.com. Is this a capacitor start motor and can I start it with a microwave capacitor I salvaged an old motor out of a air conditioner and I want to know if it's a capacitor start motor if I can start it with the capacitor I have and how to start it.  I included pictures of the motor and capacitor btw i have never worked with capacitive start motors so i need details on exactly how to wire it up with out ruining it What is the difference between "source x", ". x" and "./x" in Bash? Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$. Let $f$ be differentiable on $\Bbb{R}$ and let $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$. Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.  Attempt: If $f$ is constant we are done. Otherwise, $f$ is not constant. Suppose there is no such $a$. Then, in particular, $f$ has no extremum $\implies$ $f$ has no point where is goes from increasing to decreasing or from decreasing to increasing. $f$ is continuous since it is differentiable. Hence, since $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$, $f$ must be constant. A contradiction. Therefore either $f$ is constant and $f'(x)=0$ everywhere, or $f$ has an extremum at $x=a$ and $f'(a)=0$.   I feel like my attempt is not formal enough, any correction?  I would appreciate your help. How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k$ for some positive integer $k$?