The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $$f(n)=n+\left\lfloor\sqrt{n}\right\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$ Prove that for every natural number $m$ the following sequence contains at least one perfect square $m, f(m), f^2(m),f^3(m),\ldots$. The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times e.g $f^2 = f\circ f.$  I tried this out using induction but I got stuck and couldn't make it. Please help me with this problem.
There's a problem statement:     For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and      $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} \right\rfloor$$      for all $i \ge 0$.   Here, $\lfloor x \rfloor$ is the floor of $x$, the greatest integer less than or equal to $x$.   Hence, we have $\left\lfloor \sqrt{10} \right\rfloor=3$ and $\left\lfloor \sqrt{29} \right\rfloor=5$.      Show that for each positive integer $m$, the sequence $S_m$ contains the square of some integer.   I'm pretty certain that this can be proved with induction. I am just not quite sure what to induct on. Examining examples shows that $S_m$ always results in a perfect square eventually, though I'm not sure how to prove it.
I want to see all URLs that for packages that would be downloaded for apt install &lt;pkg&gt; without actually installing the package.  Normally, the URLs can be extracted from the screen output after installing the package (as above).  An example of a URL I want to list:    Is it only possible to get such list of URLs beforehand by looking through all the apt's package source lists? I'm trying to  when I run into this:     "You have already deleted 5 of your own posts today; further deletes are blocked"      I found conflicting information on this very annoying limit:   This post from May 25, 2010: . This is marked . If this is no longer the case, please edit that question and flag it as . Then , indicates that the problem has risen from the dead -- despite :     Each time that I've seen it (pruning subpar answers), it's been a user who is deleting their own redundant zero-upvote answers. Can you tweak the algorithm slightly so answers with less than one upvote don't count towards tripping this flag?     (Emphasis added.) :     I don't mind it when         cleaning up redundant answers on questions that have multiple alternate good answers   cleaning up old, "less awesome" answers that are no longer reflective of the user's current skill level         (Emphasis added.)     So why do we have barriers in place, to making SO answers less, "less awesome"?      Proposed Solution:  Please let us (¿once again?) delete an unlimited number of our own answers, when they have zero up-votes and are not marked as the answer.  Otherwise, please edit the linked questions to clear up the confusion. How to move the center of scaling Hello. I'm a beginner.  In the model shown, when I select all of the top vertices and then scale them to the center, the center of scaling is not the center of the 3d-model.   How do I move the center of scaling? 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How do I create a fishnet shapefile using QGIS?   The equivalent task of "Create Fishnet" in ArcGIS. I am trying to understand the sentence  the mean minimizes the mean squared error.   from wikipedia .  From a previous post  I can see a formal proof which according to the authors prove that the mean minimaze. But my objection on that prove is that is you substitute m with median all the reasoning still stands.  Therefore, I restricted myself to one dimension and built the following example :   MSE = 1/n[SUM(Y_hat_i - Y_i)**2]   Given the following distribution Y = [5,3,2,7,4], minimizing MSE I intended as in the following: I should find a function Y_hat such that when it is applied to the MSE formula (more specifically to each element of the Y distribution), we are sure that we obtain the minimum value. For simplicity, I could consider the Y_hat as any measure or function for the central tendency, including, mean, median, mode, etc.  Lets suppose, Y_hat = mean (=4.2 for our distribution), therefore, I did all the calculations such as:  1/5 [(4.2-5)**2 + (4.2-3)**2+ (4.2-2)**2 + (4.2-7)**2  + (4.2-4)**2 ] = 2.96   Let's suppose, Y_hat = median (=4 for our distribution), therefore, I did all the calculations such as:  1/5 [(4-5)**2 + (4-3)**2+ (4-2)**2 + (4-7)**2  + (4-4)**2 ] = 2.8   If I did all the calculation correctly, the result shows that for this case, the median is minimizing better than the mean.  I am sure, I am doing something wrong and missing important aspects of the overall reasoning.  Please, could you provide any clarification on the above topic? I would really appreciate it.  Many Thanks in advance, Best Regards, Carlo Joined pipelines Considering a routine such as this one:  alpha() { echo a b c |tr ' ' '\n'; }   which outputs a stream, I would like to take the output stream, transform it, and paste it with the original output stream.  If I take use upcasing as a sample transformation, I can achieve what I want with:  $ mkfifo p1 p2 $ alpha | tee p1 &gt;( tr a-z A-Z &gt; p2) &gt;/dev/null &amp; $ paste p1 p2 a       A b       B c       C   My question is, is there a better way to do this, preferably one not involving named pipes? PostgreSQL: running count of rows for a query 'by minute' Fighter War Magic, Using Booming Blade / GFB If my Fighter (Eldritch Knight) use War Magic (PHB p. 75) to use the cantrip Booming Blade, will I be able to use my Extra Attack, and bonus action attack, to make 3 attacks at level 7? How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon?   I proved geometrically that $\sin x&lt;x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin x_2|&lt;|x_1|+|x_2|$$   But this doesn't help me much finding a delta...   Thanks for any help!  P.S. I'm only at the beginning of calculus so I can't use many theorems and derivation  (because they haven't been regorously proven). Stop the Ubuntu updater from nagging about Firefox English Language pack even after uninstalling Firefox I have run sudo apt purge firefox, and I have gone through all of the /.config files, etc., and Software Updater is still asking me daily if I want to install the &quot;English Language Pack for Firefox&quot;. I don't have Firefox installed. How can I stop the updater from asking to update something on a program that doesn't exist? Output of dpkg -l | grep firefox: ii  firefox-locale-en                          88.0+build2-0ubuntu0.18.04.2        amd64        English language pack for Firefox Is there a difference in meaning among 'should', 'happens to', or 'should happen to'? Is it possible for moderators to review more than the daily limited count? How do I access a field value for an entity (e.g. node) object? node_load((int) $nid) loads a fully-populated node entity. $node-&gt;title-&gt;value gets the title, but how do I access field values in Drupal 8? I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post:   So I got the information, that the vector $|x\rangle$ form a dirac-normalized basis for the Hilbert Space.   I know that the dirac-delta distribution is defined like this: $$\delta(x-x') = \begin{cases} 0 &amp;\mbox{if } x\neq x' \\ \infty &amp; \mbox{if } x=x' \end{cases},$$ this means that my x' is a point on my x axis where I have my infinite high peak. And also $$\int_{-\infty}^{\infty}dx\cdot \delta(x-x')=1.$$  But how actually correlate this with the scalar product of vectors x, x' in the Hilbert Space that form a so-called 'diracl-normailzed" basis of it?  Can you give me some tips on this please? Or maybe you actually know a link, where this is explained. When there are too many linked questions to fit in the sidebar, it shows a link to &quot;see more linked questions...&quot;. Clicking on this link takes you to a front-page type showing only the linked questions. You can then use the tabs to sort by &quot;active&quot; to show the most recently active linked questions. However, the gray popup that appears when new questions are posted shows up for any recent activity, not just activity on the linked questions. Clicking through then updates the list, appending more questions to the top of the list, regardless of whether or not they're linked to the question at hand. To reproduce:  Go to  Wait for activity  Screenshots:   Now obviously this isn't a show-stopper, but I thought it might be worth mentioning.  Update June 2020 This is still relevant.  While sitting on the &quot;linked questions&quot; page for a specific question - in the above screenshot,  - and sorted by newest, when a new question is asked the &quot;1 question with new activity&quot; banner will show up, even if that new question does not link back to the question that you were on. Change the URL in the browser without loading the new page using JavaScript