I want a proof of the identity $${n \choose n} + {n+1 \choose n} + {n+2\choose n} +\cdots + {N \choose n} = {N+1 \choose n+1}$$ where $N\geq n$, by algebraic or combinatoric methods.
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.
kidle_inject causing very high load Whenever kidle_inject starts running on my system the load shoots up. In general it only gets to a load of around this 5.23, 3.65, 2.54 but a lot of the time it jumps up past the 15 mark on each of them. At that point the system grinds to a halt and I have to power it off.  This only happens when the kidle_inject process starts. How can I disable this process so my machine can work and I can get some work done.  Example top:  top - 12:43:48 up  3:02,  3 users,  load average: 6.57, 5.53, 3.33 Tasks: 256 total,   3 running, 253 sleeping,   0 stopped,   0 zombie %Cpu(s):  5.9 us, 16.6 sy,  0.0 ni, 74.7 id,  0.5 wa,  0.0 hi,  2.3 si,  0.0 st KiB Mem:   8080252 total,  7937856 used,   142396 free,   258904 buffers KiB Swap:        0 total,        0 used,        0 free.  4061336 cached Mem    PID USER      PR  NI    VIRT    RES    SHR S  %CPU %MEM     TIME+ COMMAND 13509 root     -51   0       0      0      0 S  54.7  0.0   3:29.47 kidle_inject/1 13511 root     -51   0       0      0      0 S  54.1  0.0   3:30.92 kidle_inject/3 13510 root     -51   0       0      0      0 S  48.8  0.0   3:11.60 kidle_inject/2 13508 root     -51   0       0      0      0 S  47.6  0.0   3:08.47 kidle_inject/0 How can I rename many files at once? What is the cause of this sudden drop in CV queue size? Why does getopts only work the first time? How can I change the class of an HTML element in response to an onclick or any other events using JavaScript? JQuery to load Javascript file dynamically Can an integer of the form $4n+3$ written as a sum of two squares? Is it acceptable to write a thank you in a comment? There have been a number of cases where I would like to thank someone for providing me with a thorough response and give them feedback beyond the vote up. Often I wish people would tell me why they voted me up. Not that the thank you part is necessary but it sets it apart from the other comments related to answering the question itself.  In this type of situation is a "Thank you" acceptable in a comment?     For more information, see  in the . Hide UIToolbar UIBarButtonItems When I studied Bayesian statistics, a question about the notation of Bayes' Theorem came to my mind. Below is the density function version of Bayes' Theorem, where $y$ is data vector and $\theta$ is the parameter vector:  $$ p(\theta|y)=\frac{p(y|\theta)p(\theta)}{p(y)} $$ The numerator on the right handside can be written as: $$ p(y,\theta) $$ which is the joint probability distribution of $y$ and $\theta$, then Bayes' theorem could be written as: $$ p(\theta|y)=\frac{p(y,\theta)}{p(y)} $$ Furthermore, $$  p(\theta|y)\propto p(y,\theta) $$ Am I right on this? I think it does not look right. Because the posterior is the proportional to the joint density function. But where is the mistake? If A and B are correlated with C, why are A and B not necessarily correlated? How can I remove snaps without losing GUI? I want to know how to check how many snaps programs/apps I'm running. Also, how to replace them (if possible) without losing GUI? I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is orthocentre and $I$ is incentre). I could find  $AP=2R\cos A$, $AI=4R\sin\frac{B}{2}\sin\frac{C}{2} $ and $\angle PAI=\angle \frac{B-C}{2}$. So applying cosine rule I got $$PI^2=4R^2+16R^2\sin^2\frac{B}{2}\sin^2\frac{C}{2} -16R^2\cos A\sin\frac{B}{2}\sin\frac{C}{2}\Bigg(\cos\frac{B}{2}\cos\frac{C}{2}+\sin\frac{B}{2}\sin\frac{C}{2}\Bigg)$$ How to proceed form here ? What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$ How to do the port forwarding from one ip to another ip in same network? I would like do some NAT in iptables. So that, all the packets coming to 192.168.12.87 and port 80 will be forwarded to 192.168.12.77 port 80.  How to do this with iptables?  Or   Any other ways to achieve the same? consider $r&gt;0 , p&gt;1$ and $K \subset B(x_0 , 2r) \subset R^n$ . $K$ compact. Define the sets :  $$A = \{ u \in C^{\infty}_{0} (B(x_0 , 2r)); \textit{ such  that }  \  u=1 \textit{ in a open  neighborhood of K} \}$$  $$ B =  \{  u \in  H^{1,p}_{0} (B(x_0 , 2r)) \cap  C (B(x_0 , 2r))  : u=1 \  on \  K  ,  \  0 \leq u \leq 1\}$$  I want to show that   $$ \displaystyle\inf_{u \in A} \displaystyle\int_{B(x_0 , 2r)} |\nabla u|^p  \ dx= \displaystyle\inf_{u \in B} \displaystyle\int_{B(x_0 , 2r)} |\nabla u|^p   \ dx$$  For this i am trying to aproximate (in the sense of the  norm of $H^{1,p}$) the function of B , by function of $A$.  Somenone can give me a hint ?  thanks in advance TikZ/PGF 3.0 datavisualization vs pgfplots To follow-up on a previous question , the current release of TikZ 3.0 has a flexible datavisualization engine that creates the same kind of plots as the more mature plotting package pgfplots.  What is the current recommendation for producing complete plots in LaTeX? How do I use the Cherokee keyboard layout to type English characters? Wow, I've really done it this time. I've locked myself out of my machine.  In preparing for a Cherokee language course, I added the Cherokee keyboard layout to my options. However, I was unable to generate any Cherokee characters by selecting that input device.  I found an article on the web which instructed me to use the dpkg-reconfigure keyboard-configuration command in a terminal.  Once I did that, I rebooted as instructed. Once I was presented with the login screen, I entered my password as usual only for it to fail.  After failing a few times, I gave my attention to what is usually a mundane exercise (i.e., logging in).  While I carefully entered my password, each time, I get the message, "Sorry that didn't work..."  So, I brought up the onscreen keyboard... But the password box displays an asterisk for only the first character of my password.  I attempted to use the direct Unicode input method (ctrl-shift-Uxxxx) but the Cherokee keyboard has no "U" key.  Can anyone please help me troubleshoot this? Check for column name in a SqlDataReader object Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers. Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers.   This is question 5 from Herstein Page 35. I have proved that conclusion holds for three positive integers.Thanks.