How to establish the formula for this without induction?  $$1 + 2 + 3 + · · · + n =\sum_{j=1}^n j=\frac{n(n + 1)}{2}$$
I am trying to compile a list of the different ways to come up with the closed form for the sum in the title. So far I have the famous Gauss story, the argument by counting doubletons, and using generating functions. Is there any other (significantly) different way to discover the closed form?  To clarify, Gauss's method was to take $1+2+3+\dots+n$, write it backwards, and add it up.
I'd like to make commands I've typed (input) into terminal stand out from all the output.   For example:  imac:~ buster$ chmod -R g-w myfolder  imac:~ buster$ cd myfolder imac:myfolder buster$ ls -l total 0 drwxr-xr-x   9 root  admin   306 Apr 20  2010 bin drwxr-xr-x   7 root  admin   238 Apr 20  2010 include drwxr-xr-x  73 root  admin  2482 May 18 17:16 lib drwxr-xr-x   6 root  admin   204 Apr 20  2010 man imac:myfolder buster$ echo Go Giants! Go Giants!  bold jumps to mind but I'd accept a color or even highlighting the whole line...   I'm sure there's a way to do this but it's not obvious to me...  thanks! Theoretical proof of the constant of speed of light $c$ in vacuum in all frames of references Is there any theoretical proof for the constant speed of light in all frames of references? I know it is experimentally proven but just curious. Hyperlinks are not visible in the generated PDF CV Is floating point math broken? I want to install Windows XP on my desktop computer (only XP), but my CD drive is not working.  I've only found installation tutorials on how to install it on the pen drive rather than from the pen drive.  How can I install Windows XP from a USB flash drive? I have 2 ADS1299 analog to digital converters(). I need to give them bipolar mode supplying but I must use a powerbank of 5V 3A. I wonder how can I create a bipolar supply of +2.5v -2.5v as mention in the datasheet from the 5V of the powerbank. I am new in this field. Thank you "For someone used to the tiny creatures we get in England it was something of a shock."  I think, in this sentence, relative pronouns before some words have been omitted. I know rules of omitting relative pronouns, but in this case I have failed to understand before which words relative pronouns can be placed to understand the sentence perfectly, which has been stated above. Please, explain the rules of omitting relative pronouns with the help of the sentence stated above, so that I can be able to understand this type of sentences easily. You can also use other sentences to explain this fact clearly.  Thanks to everyone. Trouble with gradient intuition How can I abort a running JDBC transaction? Is there a way to prematurely abort a transaction? Say, I have sent a command to the database which runs five minutes and after four, I want to abort it.  Does JDBC define a way to send a "stop whatever you are doing on this connection" signal to the DB? It is a common assumption (and, as far as historical stock market data goes, also proven) that the stock market is always growing if you enlarge the time window long enough . In other words: The stock market always goes up in the long term. Is, therefore, the conclusion valid that you can totally invest as much as you want at any point in time you want and you can never go wrong, if  You diversify strong enough (invest &quot;in the world&quot;) You do not need to urgently sell your assets at any point in time  If yes, all that strategies that rely also on diversified, long-term investments (Dollar-Cost-Averaging &amp; Rebalancing, .. what others do you know?) are just to optimize the yiels, but you'd make profit even without those strategies. Is that correct? Which Harry Potter works are considered canon? Is my Geoserver "Direct WMS Integration" working? Where Can I Give Feedback on Ubuntu? I'm running Ubuntu 11.04 Beta 2. Is there a way, or a place, to give feedback on this system? Not file bugs, just give feedback. Is this the correct use of the question mark?  Should it be at the end outside of the closing bracket?  See example below     How do we deliver it? (to the learner) We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on? Force JavaScript exception/error when reading an undefined object property? How do I make a floor plan with specific dimensions? This is my very first time of using blender. I've tried several tutorials but can't find anything that tells me how to do the very basic thing i want to do.  I simply want to draw a floor plan for a room which is 13 feet by 11 feet and make it 3d so that ultimately i can play with where things go in that room.  Can anyone take me through the steps needed to do this?   I will be so grateful.  Thank you in advance Setting print layout to portrait using QGIS 3? Is it possible to export the equations from Mathematica to MATLAB? Is a semigroup with unique right identity and left inverse a group? We know that a semigroup with a right identity and right inverse for all elements is a group (e.g. see ). Symmetrically, also a left identity together with a left inverse implies a group. We also know that a semigroup with a right identity and a left inverse is NOT necessarily a group (see ). My questions are:   in a semigroup, is the existence of a UNIQUE right identity together with the existence of a left inverse enough to have a group? in a semigroup, is the existence of a right identity together with the existence of a UNIQUE left inverse enough to have a group?   I think both these claims are false,  ut haven't found a counter-example so far.