$$\sum_{\ell=k}^n \binom{\ell}{k} =  \binom{n+1}{k+1}$$  Can someone explain how we got that equation? Expanding it we get   $$\binom{k}{k} + \binom{k+1}{k} + \binom{k+2}{k} + .....+ \binom{n}{k}$$  but how does that add up to  $$\binom{n+1}{k+1}$$
After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
Blender is rendering an older version my project @class vs. #import It is to my understanding that one should use a forward-class declaration in the event ClassA needs to include a ClassB header, and ClassB needs to include a ClassA header to avoid any circular inclusions. I also understand that an #import is a simple ifndef so that an include only happens once.  My inquiry is this: When does one use #import and when does one use @class? Sometimes if I use a @class declaration, I see a common compiler warning such as the following:     warning: receiver 'FooController' is a forward class and corresponding @interface may not exist.   Would really love to understand this, versus just removing the @class forward-declaration and throwing an #import in to silence the warnings the compiler is giving me. What are the different endings in Dishonored? Russian SE has a slightly longer and more storied history than Japanese SO I want to Setup GIMP 2.10 on Ubuntu 16.04 (Xenial Xerus) without flatpak.  can it be possible now or on future?? I have an email account that has passed 60GB of emails, and currently I'm having a lot of trouble using an email client to archive emails of last year (2011).  Via terminal, I am trying to use  to locate the files between 2011-01-01 and 2011-12-31, but no avail.  How can I find files between two dates?  If relevant, the end goal will be a batch that will move each file found, matching the date interval, to a folder. VAR or VECM for a mix of stationary and nonstationary variables I have 4 time series. One of them is stationary and rest of them are not. I need to find relation between them. I will use AIC to decide lag length.   Should I use VAR or VECM to find relation between them?  Will VAR or VECM give me relation in terms of equation which can be used for forecasting?  Do I need to perform Johansen's test of cointegration?  What good would it do? Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible It's not part of the artwork and doesn't seem to be hurting anything, but I get this on a lot of client logos and I'm wondering how to get rid of it.   I think it's part of the artboard, but can't find anything to select/change/remove it. The question  only deals with flights in the US sourced from the  . Is there some similar source for EU, Asia etc? I have accounts in Stack Overflow, Programmers, Unix &amp; Linux and Area 51. I don't want to continue my Area 51 account, so I planned to delete my Area 51 account. I have performed the following steps to delete my account, but I couldn't!  Case 1     If your account has votes or posts do the following steps:         Edit the "About Me" section of your profile to say “please delete me”.   Contact the SE support to request deletion.      I edited my "About Me" section to "please delete me". When I contacted SE Support to delete my account, I couldn't make a request. I am getting the following error:  This operation is incomplete. please do it in main site.   For deleting Area 51 account, when I do it main site, in link for deletion column I gave the Area 51 link. It says that:  it is not valid link!   Case 2     If the account has not interacted with the site yet (votes or posts), click on the delete button on your user page next to the edit link.   I am not eligible for this case!  Then how to delete my Area 51 account? Any help is appreciated! How do I escape the wildcard/asterisk character in bash? For example:  me$ FOO="BAR * BAR" me$ echo $FOO BAR file1 file2 file3 file4 BAR   and using the \ escape character:  me$ FOO="BAR \* BAR" me$ echo $FOO BAR \* BAR   I'm obviously doing something stupid.  How do I get the output BAR * BAR? Feature request - links underlined Find the infinite sum $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $ I am facing a problem, where I have to find the partial sum of a sequence/sum and with that, the infinite sum of the sequence.  $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $   The problem here is that I don't know how to proceed. I would be thankful if someone would steer me in the right direction. Size commands do not work in mathmode MWE:  \documentclass{article} \begin{document} $\wedge{\small\wedge}{\tiny\wedge}$ \end{document}   All the \wedges have the same size... Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$ Individual community preferences vs. SE network policy: who wins? In the recent question , the OP asks where the following question would be appropriate:     What would be the best way to approach someone that works for me about bad odor?   CHAOS team member Abby Miller  by suggesting the proposal  &#8212; which is in commitment as I'm writing this &#8212; because     One of the up voted example questions there is about dealing with an irritating coworker. This question will be perfectly on topic for that site.   In my opinion,  qualifies for the not constructive close reason*, which applies to all sites in the network, and therefore, the question shouldn't be allowed, no matter how much it has to do with workplaces.  I know Abby isn't the only one who disagrees with me about this. I once had a talk with a mod of some beta about the presence of poll/GTKY questions on his site. His position, in short, was that his community had different rules from SO and its members enjoyed such questions. I don't remember which site it was, it might even be graduated now, but I never did end up participating there.  Anecdotal examples are nice and all, but here's the real question: to what extent can individual communities/sites override network policies? To what extend should they be able to do so?  *: not constructive comes with the description "This question is not a good fit to our Q&amp;A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion." Title tag different from title appearing in Google? I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? How to find required information in man The man command brings up a nice manual for many programs, but how can I use it more effectively? For example man gcc brings:  NAME        gcc - GNU project C and C++ compiler  SYNOPSIS        gcc [-c|-S|-E] [-std=standard]            [-g] [-pg] [-Olevel]            [-Wwarn...] [-Wpedantic]            [-Idir...] [-Ldir...]            [-Dmacro[=defn]...] [-Umacro]            [-foption...] [-mmachine-option...]            [-o outfile] [@file] infile...         Only the most useful options are listed here; see below for the        remainder.  g++ accepts mostly the same options as gcc. .... many text   But what if I need only some part of it? What is the best way to jump to the section where -Idir is described, for example? Calculating roundness/compactness of polygon? I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?)