How do I derive the formula for the sum of squares?
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
How to avoid being prompted for a password by sudo? I know I can become root (super user) via the su command but I have to authorize it after entering the commands. Is there a way I can become root and authorize (with password) in one line ISO image integrity check (new try)? When you open an .iso file with e.g. 7-zip, the tool lets you test the ISO file, just like you can test any RAR or 7Z or ZIP file that the file is still ok.  If that test finishes successfully, does that mean the .iso file data and all the files within the ISO disc image are fine and not corrupted (bitrot)?   I am a bit confused because elsewhere I saw it mentioned that the ISO (raw disc image) format doesn't contain any data integrity checksums, so I am unsure what the 7-zip tool is testing while testing an ISO disc image file. Overfull box and broken pagination with page that contains only section headings How to correctly install libpng12-0 on the Ubuntu 19.10? For  I need to install libpng12-0 package on the Ubuntu 19.10 system.  I know that it is available .  So I have decided to download this package manually and install it by hand.  On previous Ubuntu versions (such as 18.04 LTS) this trick ends with success, but here on 19.10 it fails in very strange way:   wget http://mirrors.kernel.org/ubuntu/pool/main/libp/libpng/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb sudo apt-get install ./libpng12-0_1.2.54-1ubuntu1.1_amd64.deb  0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/116 kB of archives. After this operation, 285 kB of additional disk space will be used. Get:1 /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb libpng12-0 amd64 1.2.54-1ubuntu1.1 [116 kB] Selecting previously unselected package libpng12-0:amd64. (Reading database ... 212864 files and directories currently installed.) Preparing to unpack .../libpng12-0_1.2.54-1ubuntu1.1_amd64.deb ... Unpacking libpng12-0:amd64 (1.2.54-1ubuntu1.1) ...  dpkg: error processing archive /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb (--unpack):  unable to install new version of '/lib/x86_64-linux-gnu/libpng12.so.0': No such file or directory No apport report written because the error message indicates an issue on the local system                                                                                          Errors were encountered while processing:  /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   What is wrong here? How can I get libpng12-0 installed on Ubuntu 19.10 system?    Note: the same happens on 19.04, but it is near EOL. How to correctly enable Desktop Cube in Unity 3D? I have read several questions regarding how to fix an already enabled desktop cube or the problems it had in 11.04. But in 11.10, how can I correctly enable desktop cube in Unity 3D? How to avoid the wrinkle at the poles of the UV sphere? When I create a UV sphere and add the subdivision surface modifier, the following wrinkle appears.    How can I avoid it?  I've read the article  but there seems to be no clear solution.  UPDATE 1  I've used the word 'apply' in the first sentence of the initial version of this question. It's misleading since I want to preserve the modifier without 'applying' it. Rather, what I meant is 'adding' the modifier.  UPDATE 2  I followed josh sanfelici's solution, but still there is an artifact at the pole.  Here is the screen: (A rendered view with a sun lamp)    Maybe it's the limitation of the general-purpose subdivision modifier.  BTW, when the cast modifier is added after the subdivision modifier, the sphere becomes perfect (even for my original wrinkled sphere).  But since I want to edit the resulting sphere (for example, scaling X, Y, etc) while modifier is added but not applied, it's not usable. I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.   I have already figured out that $|Z(G)| = p$, and that $G'=[G,G] \lhd G$.  Also, $|G'| = p$ or $p²$.  I suppose I'd have to prove that $G' \subset Z(G)$, but I've been trying and I have no idea how. Any hints? Thanks! After upgrading to Firefox 57 Quantum many website have rendering issues, such as empty frames, missing text or fonts, etc. I use Kubuntu 16.04 64 bit.  So far, I've tried:   Restarting Firefox in safe mode Disabling "hardware acceleration"   Wiping out my profile Purging Firefox and reinstalling Running Firefox from a clean user account   But nothing helped so far.  Any tips on how to solve this are much appreciated In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?    Thanks for the answers. Several answers basically said for irrational numbers that  A Dedekind cut is a pair of sets of rational numbers $\{L, R\}$. The set of real numbers is defined to be the set of all Dedekind cuts, where a Dedekind cut is a pair of sets of rational numbers $\{L, R\}$ which have no elements in common, and where all the elements of $L$ are less than any element of $R$.  Each Dedekind cut is a real number. This is where I have a problem - surely that can’t be correct. The set $L$ is a set of all rationals, and there must be a rational in the set $L$ that is greater than all other rationals in that set, even if we have no method of determining it. And similarly, there must be a rational in the set $R$ that is less than all other rationals in that set, even if we have no method of determining it. If every irrational number has a corresponding set $L$, then each irrational number has some such corresponding largest element of that set $L$, and then each irrational number has some corresponding rational number. And that would mean that the irrational numbers are countable. So, with Dedekind cuts,  the only conclusion is that there must be irrational numbers $x$ which are either greater or lesser than some irrational cut $y$ of the rationals, and between $x$ and $y$ there is no rational number.  But that is impossible, so that the Dedekind cuts cannot be the correct representation of the real numbers.  Surely the problem with Dedekind cuts is in using sets of rationals that include all rationals up to a certain rational. But there is an alternative method of representing irrationals can be defined in terms of infinite sets of rational numbers. For example, in binary notation, the non-integer part of $\pi$ is $.00100100\ 00111111\ 01101010\ 10001$. You define a set by: if the nth digit is a $1$, then  the natural number $n$ is in the set. And then we have that, for the real numbers between $0$ and $1$, that the set of real numbers is simply the set of all subsets of natural numbers. Each subset corresponds to some real number between $0$ and $1$.  And in this way, all real numbers can be considered to be some set based only on nested sets of the empty set.  But I still haven’t got a satisfactory answer for how negative numbers can be represented in terms only of sets containing the empty set. Any ideas? One page website - Effect on SEO I'm considering re-designing my personal website to promote my web services in my local area.  My issue is whether to do a one page website with a scroll-to navigation system. What effect does this have on SEO. As i understand it, you set stuff like h1's and meta description on specific page content. How does this work when all of your different content is on one page? How do I add a new language? The aftermarket ROM I use seems to have the language list parred down to just a couple selections, probably to save space.  I would like to add a language back into the system. I can already set the locale using programs such as  so that applications with my language support start with the right locale code, however that doesn't add missing system languages. Having the system in another language would be nice, but the main thing is getting the dictionary files for the keyboard auto-complete. How does one go about adding the language support files for an input language to a system or ROM?  I can access other ROM's or the AOSP project to rip things out, but I don't know where to start. What files do I need and what modifications need to be made? Is there an easy way to do this with say a language-pack or is it hackery all the way?  My ROM is fully rooted and I have adb access if necessary. It's Android 2.3.7 () and I'm using the stock Android keyboard. The language I want is Turkish, although unless somebody has a download to point me to it's mostly the process I'm looking for here. Proof that there must exist infinitely many solutions to a homogeneous system of equations when more unknowns than equations I am sitting inside a closed room with a cement ceiling just above my head. The pressure of the atmosphere is exerted on the ceiling from above(outside). But solid cement not being a fluid does not transport that pressure into my room. But i still feel the atm pressure which has no way to be transported into the room.I do not boil to death or anything. How does this happen? sudo find / -name "*" | xargs grep -sn --color=auto "-j"   Above command returns below:  grep: invalid option -- 'j' Usage: grep [OPTION]... PATTERN [FILE]... Try `grep --help' for more information. ...   How do I search for string -j? When we measure the speed of a moving element we do it with the help of a reference frame. Now if we need to measure the speed of time, is it possible? Does time actually have a speed? Difference between staticmethod and classmethod What is the difference between a function decorated with  and one decorated with ? How can I make a hexagonal grill? How would I be able to create a grill of sorts found on the front of some computer cases such as this one Should the user be forced to sign in for a wish list on an ecommerce site? I believe that signing into a site can be a big barrier to users so prefere e-commerce sites that offer guest access.  The question is should wishlists be a guest feature or do the business advantages enforcing the user to sign up outweigh the potential increases in profits.   Does anyone have any research or views on this? To better describe the above problem, whenever I start up my Ubuntu/Windows 10 Machine and select Ubuntu, whenever the machine tries to start the GNOME Desktop Environment, it says it started it successfully. however, I never get a graphical desktop environment to use. I can fix this by logging into tty3 and typing gnome-session but I don't really want to do that as, for some reason, that skips login. Can anyone help?