Prove by induction that for all $n \in \Bbb N$   $$\sum_{k\mathop=1}^n k = \frac{n(n+1)}2 \quad \text{and} \quad \sum_{k\mathop=1}^n k^3 = \left(\sum_{k\mathop=1}^n k\right)^2$$   I have proved the first part but can anyone show me how to solve the second part?  Thanks.
How can I prove that     $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$    for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.  Thanks
How do I close a terminal without saving the history? More than once I've accidentally run a number of commands and polluted my bash history. How do I close my terminal without saving my bash history? I'm using Fedora. Namespace constant in C# Grub: reboot to specific kernel I'd like to ask for help getting grub-reboot to work.  The theory: According to  (and other sources) I should be able to choose the kernel for the next reboot, using grub-reboot. However, this does not work for me; grub always boots to the default kernel.  Here's what I tried:  $ grep GRUB_DEFAULT /etc/default/grub GRUB_DEFAULT=saved $ grep "menuentry " /boot/grub/grub.cfg  menuentry 'Ubuntu' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-simple-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-43-generic' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-43-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-43-generic (recovery mode)' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-43-generic-recovery-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-36-generic' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-36-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060' {     menuentry 'Ubuntu, with Linux 4.4.0-36-generic (recovery mode)' --class ubuntu --class gnu-linux --class gnu --class os $menuentry_id_option 'gnulinux-4.4.0-36-generic-recovery-f3a807d1-3fd6-4970-b741-5382e9801060' {  # Version 1, using the name $ sudo grub-reboot "Ubuntu, with Linux 4.4.0-36-generic"  # Version 2, using the id $ sudo grub-reboot gnulinux-4.4.0-36-generic-advanced-f3a807d1-3fd6-4970-b741-5382e9801060  # Version 3, counting $ sudo grub-reboot 3   Neither of these commands seem to have an effect; after reboot, I always end up with the 4.4.0-43 kernel.  Note: This is on Ubuntu 16.04 Xenial. It is a server machine and I don't have access to the console during boot, so I can't look at the grub menu :( JavaScript raises SyntaxError with data rendered in Jinja template How do I make a bootable Ubuntu USB? How do I make an Bootable Ubuntu USB containing the the latest version of Ubuntu?  For completeness, how would I be able to do this on Ubuntu, Windows, OS X or another Linux distro? Does white whole wheat flour need to be refrigerated? I know that regular whole wheat flour needs to be refrigerated, but does white whole wheat flour also need refrigeration? I hate to use up the space in the refrigerator if I don't have to.  I have 5 lb. (2.3 kg), and it would probably take me a month or so to use it all. "All of the ...." or " All the ... "? Is "funnest" a word? We seem to be stuck at an impasse on this issue.  Is funnest a word or not?  If so, does it mean "most fun"? Manipulate runs fine normally, but gives errors as a CDF I am looking for an adjective, similar to "holding a grudge" I am looking for an adjective that describes a person who is not willing to forgive or excuse someone's faults or wrong-doings from the past (other than "unforgiving"). For the ones who speak German, that word would be "nachtragend". I bought a used iPhone 4.  The previous owner re-set the phone, but did not remove the device from their iCloud.  It is asking me for their iCloud information when I try to set-up the phone.  I don't have contact with the previous owner anymore.  Is there a way to remove the device from their iCloud so I can use a new one? How can there be a directory named `~`? I was trying to test out this buggy/new tool to see how it works, and it prompted me to choose a directory to use for stuff I am doing. From the directory ~/Code/indep, I typed in ~/Code/indep/tarbell. (Stupid, I could have just typed 'tarbell', but whatever.) It prompted me to create it because it didn't exist, I said yes. When I inspected the contents of this directory, here is what I see:  brian@navis:~/Code/indep$ ls ~ project1 project2 project3 ...   I thought "oh, weird, it created a symlink to ~". However:  brian@navis:~/Code/indep$ file ./~ ./~: directory   file thinks it's a directory.   brian@navis:~/Code/indep$ ls -l drwxr-xr-x 3 brian brian 4096 Nov  8 16:50 ~ drwxr-xr-x 8 brian brian 4096 Oct 14 18:42 project1 drwxr-xr-x 3 brian brian 4096 Sep 10 21:15 project2 drwxr-xr-x 5 brian brian 4096 Oct 14 19:33 project3 ...   And so does ls -l.   I thought, "oh, ... is it a hard link? I've never really used those." But I tried to reproduce, and it's impossible to create a hard link to a directory. So... I guess I've narrowed this down to things I don't understand. Did I mount my home directory in this other directory somehow?   My main goal at this point is to remove the reference to my home directory in this folder (rm doesn't seem like the answer), but I'm also curious to know what's going on.  EDIT:   In retrospect, I understand exactly what happened. As was answered below, the program created a directory called "~" inside my current directory. The only really obfuscating fact about this is that I wasn't consistent with how I was actually viewing what the directory was. Sometimes I referred to it as  "~" and sometimes as "./~". It's only if you do the latter that you can actually see what's going on, but until I understood this, I was conflating what I was seeing from using the absolute path of "~" to to what was in fact a local directory. Prove, in this figure, that $EFGH$ is a parallelogram I know this is a very common corollary of the class equation. And I know how to do it by using class equation.   But can you do it bu using group action, maybe find a nice set for G to act on? I have one site with two languages: English and Spanish for example. What would be the proper convention to name image files?  From my point of view it would be stupid to duplicate image and then name it according to on which site it will be uploaded. It would be waste of resources IMO and unnecessary job to do.  What do you think about this one? UAE visit visa with expired passport number I need a visa to travel to UAE. My passport will expire soon (June), so today I applied for a new passport and it will be delivered to me on March 17. I need to travel to Dubai on March 18. clearly not enough time to get a visa for the new passport.  Can I send a scan of my old passport to the agency who's getting the visa for me, get a visa with the old passport number on it and then travel with both passports to Dubai? My new passport to satisfy their 6 months validity requirement and the old passport to match the passport number? Markdown change: Intra-word emphasis now works There are tons of questions I could maybe post this as an answer to, but before choosing the best one, I'll just post this as an announcement question that can be linked elsewhere. The short story From now on, intra-word emphasis is possible in our Markdown version. That means you can now write  The &quot;V&quot; is for Model**View**Controller. That's un*fricking*believable.  and it renders like this:  The &quot;V&quot; is for ModelViewController. That's unfrickingbelievable.  This only works for asterisks, not for underscores. It works in posts and comments. The reasons There are two big reasons for this. For one thing, this is more or less the behavior of , which we eventually . &quot;More or less&quot; because CommonMark's emphasis behavior is still in flux, and supports a few things that we don't support yet, but at least with this change we're closer. The other reason is languages that don't even have the concept of &quot;words separated by spaces&quot;. See  and  for two examples. This change allow posts in those languages to emphasize words without having to resort to HTML tags. To use an example from the first linked post,  **北京**和**上海**是直辖市。   now renders like this:  北京和上海是直辖市。  Possible issues In the 99% cases, this should only make things better; I've only found very few posts where this change causes unintended bolding/italicizing. And even in those cases, note that the rendered versions of questions and answers are not re-rendered until a post is edited, so this will not affect anything in posts retroactively. Unlike posts, comments are rendered , so this immediately applies to all comments. In most cases this doesn't change anything or actually fixes things. Should you discover a rare case where this breaks an important comment badly, we can always edit it to fix. There's one side effect that I'd like to mention: If you find a great quicksort implementation on GitHub that has a bug that causes a crash when sorting Windows versions, and the author doesn't want to fix it, you may be inclined to say  f**k this s**t   (i.e., &quot;fork this sort&quot;). With intraword emphasis, this would however look like this:  fk this st  So you would have to escape the asterisks (one of them would actually be enough). I'm not worried about this case too much, since if you must put censor-worthy words into a post, you might as well have to jump through a little hoop. Some details This being Meta, there are probably lots of people interested in a more detailed description of the new behavior (don't feel bad if you're not one of them; just skip this section). Here are the major rules regarding intraword-emphasis:  As mentioned, it only applies to asterisks, not underscores. Underscores  this restriction was in place at all; asterisks are much less of a problem.  If a single or double asterisk is supposed to be interpreted as intra-word emphasis, it cannot have punctuation on one side; it actually must be inside a word. An example of an edge case this solves is  It's a web*-based solution. *see Wikipedia for a definition of &quot;web&quot;  where the asterisks after &quot;web&quot; could cause unintended italicizing if there was a second asterisk in the same paragraph somewhere.  For italics, the &quot;inside a word&quot; part of rule 2 is even a bit stricter. For bolding, any word or digit character is allowed; for italics, digits don't work either. That's because the asterisk is often used as a multiplication symbol, so  The game worked in 320200 and 640480 resolution.  wouldn't look right without this restriction.  A minor addition that I made that's only tangentially related to intra-word emphasis is that a sequence of more than three underscores or more than three asterisks will no longer take part in any emphasis. This shouldn't really cause any problems, but fixes some (e.g. the underline in the second blockquote in , while not really visible, is a weird combination of bold and italic underscore characters; after this change, it would just be (more) underscore characters). This behavior also matches CommonMark. How can I convert string to datetime with format specification in JavaScript? FFmpeg lossy compression problems Getting computer's UTC offset in Python