Let's say we start with the number one, then we add $(1/1)$. Now we have $2$. Then we do $2 + (1/2)$. Now we have $2.5$. Then we do $2.5$ plus $(1/2.5)$. If we continue this forever, I am fairly sure this diverges. However, I am wondering whether there is an implicit formula for the $n$th term of this sequence. For example: $A(1)=1$, $A(2)=2$, $A(3)=2.5$, and so on.
Today, we had a math class, where we had to show, that $a_{100} &gt; 14$ for  $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$  Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is?
Dragon souls not activating new words So if I understand correctly you can learn new words and the souls will activate these words. I just killed a dragon and when it died it looked like I absorbed the soul but it didn't unlock any of the words I have.   I noticed some dragons actually have a name so do certain dragons unlock certain words? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know. Would it be possible to generate the original data from a SHA-512 checksum? So, I have a file that is XY Exabytes big. I create a SHA-512 hash from it.    Are there any theoretical chances that I can retrieve the original data just from the sha512 hash?    Also, are there any better hashes then sha512? I recently saw a question titled "" (10k+ users only). I thought, "surely someone has asked this question before", and began looking for a duplicate.  is a question with the exact same title, just with a little different punctuation and casing. It was "Jquery vs Javascript" and "jQuery vs. Javascript?". The only difference being the ., ?, and the casing.  Since we now , I propose that when checking for questions with duplicate names, certain punctuation be striped out to prevent the duplicate from occurring in the first place.  As pointed out in the comments, if isn't already, white-space and non-printable characters may want to be removed for duplicate detection as well. One-sided Fisher's exact test and its complement Positrons versus holes as positive charge carriers : [The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.]  and:  [Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.]  It always confused me to think of holes as positive charge carriers in semi-conductors as not being real: real electrons move from one lattice-position to another lattice-position, which effectively looks like a positive hole in the lattice that is moving in the other direction, but in reality a real electron moves, the hole is kind of an "illusion".  On the other hand the positrons are always introduced as real hard-core particles.   The quotes from the Wikipedia article make me unsure: how should I look upon these phenomena?   Edit: holes in a Dirac sea give rise to real pos. entities in one case and to unreal pos. entities in another - how can we distinguish, is it a matter of formalism? Is there any simple method to calculate $\sqrt x$ without using logarithm Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using logarithm? More simple techniques would be nice.  Here is a simple technique used to approximate square roots by :  For example: $\sqrt {78}\approx 8\frac { 14 }{ 17 } $ , where $8$ is the nearest integer root of $78$, $14 = 78 - 8^2$, $17 = 2 \times 8 + 1$.  if $n=2^k$ we can use the method above.   For example, for $k=2$ Lets calculate $\sqrt [ 4 ]{ 136 } $: $$\sqrt [ 4 ]{ 136 } =\sqrt { \sqrt { 136 }  } \approx \sqrt { 11\frac { 136-{ 11 }^{ 2 } }{ 11\times 2+1 }  } =\sqrt { 11\frac { 15 }{ 23 }  } \\ \sqrt { 11\frac { 15 }{ 23 }  } \approx 3\frac { 11\frac { 15 }{ 23 } -{ 3 }^{ 2 } }{ 3\times 2+1 } =\frac { 544 }{ 161 } =3.38\\$$ The exact result is$$ \sqrt [ 4 ]{ 136 } =3.4149\cdots$$ The method approximates well, but it is working for only $n=2^k$ as I know. Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? Legendre symbol: Showing that $\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0$ I have a question about Legendre symbol.  Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies: $$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$  I know that $\displaystyle\sum_{m=0}^{p-1} \left(\frac{m}{p}\right)=0$, but how do I connect this with the previous formula?  Any help is appreciated. Proof on Catalan numbers NULL values inside NOT IN clause This issue came up when I got different records counts for what I thought were identical queries one using a not in where constraint and the other a left join. The table in the not in constraint had one null value (bad data) which caused that query to return a count of 0 records. I sort of understand why but I could use some help fully grasping the concept.  To state it simply, why does query A return a result but B doesn't?  A: select 'true' where 3 in (1, 2, 3, null) B: select 'true' where 3 not in (1, 2, null)   This was on SQL Server 2005. I also found that calling set ansi_nulls off causes B to return a result. Strange Pop-up appldnld.apple.com:80 I received a strange Pop Up message. The unsettling part is password will be sent in clear text.  Is my phone affected with any malware or app virus ? How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:  $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$  There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2  a + (n-1)\cdot d}{2}\biggr)$$   How do we prove the above two identities? What’s the difference between "Array()" and "[]" while declaring a JavaScript array? Terminal flashes on empty backspace with Yosemite Since using Yosemite, I've had the annoying problem of my current Terminal window flashing when backspacing on an empty line. I know on previous versions of OS X this flashing was not default, and could be toggled in accessibility settings, but I cannot find the same setting and I'm dying to get this turned off. Any known ways to get rid of it? "BAD ARCHIVE MIRROR" error while installing ubuntu server 12.04 I downloaded the server version of Ubuntu from Ubuntu's official site. Created a Bootable USB from the ISO file I downloaded. I connected through Internet using DHCP in the installation process. While I was installing it on my Desktop Computer (pentium D 1 GB RAM) I have this error "Bad Archive Mirror".   I also found 2-3 questions with same problem on forum but I didn't get solution of my problem from them. Best way to list files in Java, sorted by Date Modified? I want to get a list of files in a directory, but I want to sort it such that the oldest files are first.  My solution was to call File.listFiles and just resort the list based on File.lastModified, but I was wondering if there was a better way.  Edit: My current solution, as suggested, is to use an anonymous Comparator:  File[] files = directory.listFiles();  Arrays.sort(files, new Comparator&lt;File&gt;(){     public int compare(File f1, File f2)     {         return Long.valueOf(f1.lastModified()).compareTo(f2.lastModified());     } }); Labeling Multiple Equations Per Row of a Table I am trying to create a table with two columns of independently labelled equations (1-6). Unfortunately, everything I have found so far either has not worked or would number them as sub-equations (1a, 1b, etc).  My current attempt is:  \documentclass{article} \usepackage{graphicx, verbatim, tabularx} \usepackage{amsfonts} \usepackage{amsmath, amssymb}  \newcommand{\grad}{\vec{\nabla}}       \newcommand{\prtl}[2]{\frac{\partial #1}{\partial #2}}  \newcommand{\curl}{\grad\times} \begin{document}  \begin{tabular}{p{5cm}p{5cm}}  $\curl\vec{E} = -i\omega\mu\vec{H}$ &amp;   $\curl\vec{H} = i\omega\mu\vec{E}$ \\  \begin{equation}  \label{eq:curlHx}   \prtl{E_z}{y} +\gamma E_y= -i\omega\mu H_x  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEx}    \prtl{H_z}{y} +\gamma H_y= i\omega\mu E_x  \end{equation}\\  \begin{equation}  \label{eq:curlHy}    -\prtl{E_z}{x} -\gamma E_x= -i\omega\mu H_y  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEy}    -\prtl{H_z}{x} -\gamma H_x= -i\omega\mu E_y  \end{equation}  \\  \begin{equation}  \label{eq:curlHz}   \prtl{E_y}{x}-\prtl{E_x}{y} = -i\omega\mu H_z  \end{equation}  &amp;  \begin{equation}  \label{eq:curlEz}    \prtl{H_y}{x}-\prtl{H_x}{y} = -i\omega\mu E_z  \end{equation}  \end{tabular}  \end{document}   Is there an environment other than tabular that would make this work? Tabularx came close (got the numbering correct), but the formatting/spacing was horrific. How to delete blank lines starting from line 5 I have this:  sed -i '/^$/d' temp_spec.rb   which is stripping blank lines and works well.  How can I make it only do this for lines 5-999 (or ideally 5 to end-of-file).  I tried:  sed -n5,999 -i '/^$/d' temp_spec.rb sed '5,999!d/^$/d' temp_spec.rb   but neither worked (no errors). regular expressions / pattern matching on ubuntu command line