$$ S_n = 1+2+\cdots+n=\frac{n(n+1)}{2}.$$ How do I go about solving this ? I tried to search for a solution, but I didn't find anything ( well I didn't know what this was called to even look for it in the first place).
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.  How? What's the proof? Or maybe it is self apparent just looking at the above?   PS: This problem is known as "The sum of the first $n$ positive integers".
Modulus with a negative number Is there anyway to restore the Blender default Cloth settings? Or is there anyway to find what the default settings are?  I didn't realize at the time that by changing the values I was overriding the default values.  Using Blender v2.76. Does going outside of range or line of sight after casting a spell have any effect? I am looking for a driver for the TP-Link AC600 Archer T2U Nano, on Kubuntu 18.04 (kernel version 4.15.0-51-generic).   I already tried different drivers, none of them worked. Linear congruence fill in the missing step? Currently working on this problem and I'm having trouble seeing how it goes from one line to the next.    $45x \equiv 63\mod 11$ goes to $x \equiv 8\mod 11$     Any help would be awesome thanks.  Please try to explain how it works to help me see thanks. Can't append  element Any idea why the piece of code below does not add the script element to the DOM?  var code = "&lt;script&gt;&lt;/script&gt;"; $("#someElement").append(code); Custom date format with jQuery validation plugin How can I specify a custom date formate to be validated with the  for jQuery? What resolution should a large format artwork for print be? For wall-sized graphics and large banners (e.g. 3m × 5m), what is an acceptable PPI for print? As I understand it 300 ppi is the typical for 'small' artworks (esp. for clean text resolution). However, for small artworks, the audience usually engages at a much closer distance. Therefore, in the interests of keeping a bitmap file down to a reasonable size, can you get away with a much-reduced PPI for a large format/banner print? What happened to the "very low quality" flag reason? Possible Duplicate:          I see a  to receive a "very low quality" flag by me.  It's just a link pointing elsewhere basically saying, "try this." I generally do this for answers I see that are like this and have been successful every time so far.  I save "not a real answer" flag for other cases such as someone asking another question in an answer or "thanks" answers.  I want to flag it as "VLQ" but I don't see that as an option*.  In fact, I don't see it as an option anymore on any site throughout the Stack Exchange network.  * This was not possible before when I initially posted this. Michael's observation on the needed downvote seems to be the reason it didn't show up.  Did it just get phased out as being unnecessary?  Should I be using "NARA" as the flag reason for this case now?  I'm not sure if the "NARA" explanation really covers this case well: (emphasis mine)     This was posted as an answer, but it does not answer the question. It should possibly be an edit, a comment, another question, or deleted altogether.   These sorts of answers can answer the question in general.  The site linked to might very well completely answer the question.  (in this particular case, maybe not however)  It's just that the answerer really should do something to improve the answer.  It's not like it's a "This question is a duplicate of this [onsite question] post" or some other general spam.  Those cases are clearly "NARA"/spam.  There was a  regarding the necessity of this flag reason.  Answers point out that it is needed and I agree.  There hasn't been anything mentioned (that I am aware of) in terms of removing it by the team.  Was this officially removed now in favor of other flagging reasons?  Or is this just a bug and it was "accidentally" removed? Iterate a list as pair (current, next) in Python What is Chromatic Aberration? Is it a physical part of the lens, or just an optical illusion? How can I make a PATCH HTTP Callout from Apex? I need to make a PATCH http request.  I made the rather incredible discovery today that I can't set an HttpRequest to use the PATCH method, as when sending the request I get a callout exception:     System.CalloutException "Invalid HTTP method: PATCH"   The work around I've found on Stack Exchange and elsewhere is only applicable to a Salesforce Rest API request:    This is therefore not helpful to me as I'm calling out to a third party.  By what manner of magic can I send a PATCH request from Salesforce?  The only work around I can think of is to not do it from Apex - this is not an option. In this  of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is said that the simply connectedness of Mandelbrot set is easier to prove.  However, I cannot find a proof through google. Can anyone provide me a reference or a proof of the simply connected of Mandelbrot set? What is the difference between a framework and a library? What is the difference between a framework and a library?   I always thought of a library as a set of objects and functions that focuses on solving a particular problem or a specific area of application development (i.e. database access); and a framework on the other hand as a collection of libraries centered on a particular methodology (i.e. MVC) and which covers all areas of application development. Per the kernel documentation:  This control is used to define how aggressive the kernel will swap memory pages.  Higher values will increase aggressiveness, lower values decrease the amount of swap.   However this is kind of vague. I'm trying to figure out exactly what the parameter ultimately controls. I know it adjusts how aggressively the kernel tries to swap out pages, but what decision making process in the kernel code does it affect?  Does it adjust how long ago a page must have been accessed before the kernel swaps it out? If so what does the default value of 60 represent? And how much is changed by a increment/decrement of 1 (a formula would be nice)? Or does it swap out pages based on their access frequency? Or something else? import Excel sheets by name My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540) Every open set in $\mathbb{R}$ is a countable union of closed sets Second-order non-linear ODE $2tx'-x=lnx'$  I differentiated both sides with respect to x:  $x'+2tx''=\frac {x''}{x'}$  Substituting $p=x'$,  $p+2tp'=\frac{p'}{p}$  But I have no clue what can I do from here on.  EDIT: $t$ is the non-dependent variable. How do I change the default application for unknown file types? Ever since installing Bless Hex Editor, all unknown files and text files automatically open with Bless instead of gedit. How do I change it back? I don't want to have to manually associate files with gedit, I just want gedit to be the default if there's no other application set up to open the file.  Solution: modify ~/.local/share/applications/mimeapps.list  [Default Applications]  - text/plain=bless.desktop - application/octet-stream=bless.desktop + application/octet-stream=gedit.desktop