Trying to prove by induction that  $\sum^{i=n}_{i = 0}$$n\choose i$$=2^n$  So obviously, I prove for some arbitary number, i.e $n = 2$  I then go on to show what the n = k terms look like for the first 3 terms  Then I realise to show that it is the same for $n = k+1$, I need to show that:  $2\times(n=k)\equiv (n=k+1)$  Yet when I try and work out the algebra I keep coming excruciatingly close but to no success, am I going about this proof the wrong way or? I've redone my calculations multiple times so there are no errors in my algebraic simplification.  Could it be that I need to go about this another way instead of trying to show it is $2\times n=k$
Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction.
How do I find slimes in Minecraft? In early alpha I remember Minecraft videos showing slimes deep underground. I think you would attack them then they split into several smaller slimes. But I've never found a slime for myself. In what conditions can you find slimes? Are they definitely still in Minecraft? What's the best way to season a cast iron skillet? I just purchased a new cast iron skillet. What's the best way to season it?   And if I need to re-season an old pan, is the process any different? Pros and cons of AppSettings vs applicationSettings (.NET app.config / Web.config) When developing a .NET Windows Forms Application we have the choice between those App.config tags to store our configuration values. Which one is better?  &lt;configuration&gt;    &lt;!-- Choice 1 --&gt;   &lt;appSettings&gt;     &lt;add key="RequestTimeoutInMilliseconds" value="10000"/&gt;   &lt;/appSettings&gt;    &lt;!-- Choice 2 --&gt;   &lt;configSections&gt;     &lt;sectionGroup name="applicationSettings" type="System.Configuration.ApplicationSettingsGroup, System, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c5612342342" &gt;         &lt;section name="Project1.Properties.Settings" type="System.Configuration.ClientSettingsSection, System, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c5612342342" requirePermission="false" /&gt;     &lt;/sectionGroup&gt;   &lt;/configSections&gt;   &lt;applicationSettings&gt;     &lt;Project1.Properties.Settings&gt;       &lt;setting name="TABLEA" serializeAs="String"&gt;         &lt;value&gt;TABLEA&lt;/value&gt;       &lt;/setting&gt;     &lt;/Project1.Properties.Settings&gt;   &lt;/applicationSettings&gt;  &lt;/configuration&gt; I have installed the Citrix Receiver 13.0.0.256735 as described for Ubuntu 13.10 here:   The installation has succeded under Ubuntu 14.04 64 bit. But if I start the Citrix Receiver nothing happens.  Is there someone who has it running with Ubuntu 14.04? How to provide a file download from a JSF backing bean? The current process for self-deletion of a user account has several problematic aspects, both in the process itself and in the end result. Some of these issues are:   The process requires manual intervention of a mod or SE employee, leading to delays and sometimes mistakes. It is more complicated than it has to be, which can lead to the impression that SE is imposing some rather arbitrary-looking restrictions and steps one has to take. This can give the impression that the user isn't really in control of his private data at SE. It is irreversible once the account is deleted, although an account can be partially reconstructed with a lot of manual effort from an SE developer. There is no real protection against mod/SE employee errors, if an account is deleted in error, there is no easy way to undo the damage. Because it is manual, it consumes the time of SE employees that have to deal with it  (unless  is used), removing useful voting information and annoying users that lose reputation that way.   I'm proposing an alternative way of handling self-deletion that would be automated, and also reversible to some extent, while still being pretty much a hard-delete (and not a soft-delete that would be problematic with respect to user privacy). The process would work roughly like this:   The user clicks a "delete profile" button in his profile. There should be a way to indicate if you want to delete all profiles on all SE sites, only on this site and a way to select multiple sites from a list where the profile should be deleted. It is verified that the user has true access to the account in some way (confirmation mail?) and not just someone that found the unlocked computer of the user. A countdown until deletion is started (I'd go with 48h, but that could be different). The timer shouldn't be too long, but long enough to correct mistakes and leave users a bit of time to reconsider. Alternative options would be to enable users to override the countdown if they really want to nuke their account immediately. Another option would be to soft-delete the account immediately, and make the countdown only for hard-deletion of the user data. Upon deletion, the posts of the user and his votes are transferred to a newly created blank account, the original account itself is hard-deleted with all the remaining data it contains. The dummy account that holds the user's data should be flagged in some way as deleted, so that the profile links will be disabled, the profile won't be accessible and it won't appear in any user lists.  The user receives a private code that can be used to claim ownership of the account. Other ways are not possible as all private information is removed about the account. This allows for an automatic way of reclaiming most of the important parts of a deleted account. The ability to restore a deleted account might have to be somewhat restricted to avoid misusing it to get rid of the account history. It is more meant as an additional safeguard so that account reconstruction is more easily possible in rare cases. Such additional safeguards might be necessary as the deletion itself would be automatic and without oversight from SE or moderators.   I personally think that the information which posts and votes were connected to a (now anonymized) account is not sensitive enough that it would have to be removed on account deletion. My idea is that this would be close enough to a hard-delete that it satisfies any privacy concerns, but still allow reassociation of posts with a recovered account. The partial reversibility would alleviate concerns about removing the manual oversight by mods and SE employees over the account deletion process. Motor to spin slowly, low torque Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms? Context: Rings.     Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?   Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck... Is it OK to leave "What have you tried?" comments? check if Gutenberg is currently in use How can I check if the editor that is currently being used is Gutenberg in a WordPress plugin?  I need this because Gutenberg lacks post_submitbox_misc_actions, so I need a fallback which will only be used if the current editor is Gutenberg. I have a long line of code that I want to break up among multiple lines. What do I use and what is the syntax?  For example, adding a bunch of strings,  e = 'a' + 'b' + 'c' + 'd'   and have it in two lines like this:  e = 'a' + 'b' +     'c' + 'd' I look for similar questions to this one but my problem is different. I want to join this community:    so I click in     but I got this error:    but then, I found that the community is already in list of communities:    However, if I click on the community, I'm an anonymous member     and even I have the option to join again     but if I click on it, the problem starts again :( what can I do? Is there a word for a person who avoids their responsibilities? Is it possible to create matter? Is it possible to create matter? In a recent discussion I had, it was suggested that with enough energy in the future, "particles" could be created.   It seems like this shouldn't be possible due to conservation but perhaps I could be wrong. Would any of you Physics masters care to elaborate?  (Note... I will understand the basics, I am by no means an expert.) According to  , gold badge holders can now unilaterally close questions as duplicates. I hold seven gold badges for tags on SO, which means I can, on my own, close questions as duplicates for those tagged questions.  Simply put: DO NOT WANT.  However, I do not see any ability in SO (the only SE site I spend much time in) to opt out of any privileges, let alone this one.  Is there a spot that I'm just not seeing that allows me to turn off this new privilege? How to change default action for file types? There is a certain text editor that I would like to set as the default editor for certain file types but if I try selecting it on the “Open With…” menu, that does not work. However, it has put an explorer context menu item in, which does work. How could I make this menu item the default double-click action, for certain file types?  Thank you SSRT and Josh R. Both good answers. It's a pity I can't give you both best answer.  For what it's worth, I found the answer today (twenty-four hours after asking this question). I wonder if this is a bug, or if I missed something... When a light hits tangentially convex surfaces with "smooth shading", the shadow on this surface seems to "partially ignore" the smoothness of the surface. Here is an example with a simple cylinder :    On the left, a cylinder with "flat shading". Everything's ok. On the right, the same with "smooth". We can see an edge, bad shadowed.  With or without the "Edge split" modifier, no matters, the edge is still there. The shader is a simple white diffuse shader with no texture nor displace.  I have to add a subsurf modifier with at least 2 levels, which will insanely increase the vertex count... And I can't do that for my project.  I hope someone already saw it, and have a little tip to share. Or if someone already know it's a bug, or a limit of the current Cycles renderer ? Why am I getting "line 1: $' :\r': command not found"? I use Cygwin on my laptop (DOS). I have a collection of scripts from my colleagues, and my own. I am not an IT person, not knowledgeable in Unix. I am following my colleagues' syntax and able to manage a few simple things. The scripts worked well on my old laptop. I just changed laptop and installed Cygwin. When I run my scripts, they do not work. Here is one example of the error message I get: line 1: $':\r': command not found line 5: syntax error near unexpected token `$'\r'' line 5: `fi  Here are the top 5 lines of my script : iter=1 if [ -f iter.txt ]    then rm ./iter.txt  fi    Can someone please explain how I can get around this problem? How can I left-align a caption? I am using this for captions:  \usepackage[font=small,format=plain,labelfont=bf,up,textfont=normal,up]{caption}   But they all come out centred - I want them aligned with the left margin - how can I do this? Solving quadratic equations in modular arithmetic Is there a general way to solve quadratic equations modulo n? I know how to use Legendre and Jacobi symbols to tell me if there's a solution, but I don't know how to get a solution without resorting to guesswork. Some examples of my problems:  $x^2 = 8$ mod 2009 $x^2 + 3x + 1 = 0$ mod 13  Thanks