Use mathematical induction to prove that $\forall n\in \mathbb{N}$, $$\sum ^{n}_{i=1}i^{2}=\dfrac {n\left( n+1\right) \left(2n +1\right) }{6}$$
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other?
Can you trip as part of a Tome of Battle Maneuver? If a ToB Maneuver such as Daunting Strike calls for a melee Attack as part of the maneuver, can this melee attack be resolved as a trip attack/include a trip attempt? single email limit exceeded salesforce very frequently..? Running junit tests in parallel in a Maven build? I'm using JUnit 4.4 and Maven and I have a large number of long-running integration tests.  When it comes to parallelizing test suites there are a few solutions that allow me to run each test method in a single test-class in parallel. But all of these require that I change the tests in one way or another.  I really think it would be a much cleaner solution to run X different test classes in X threads in parallel. I have hundreds of tests so I don't really care about threading individual test-classes.  Is there any way to do this? Does the existence (now proved) of gravitational waves imply the existence of Gravitons? I studied the theoretical part about the Gravitational waves in General Relativity (linearization of gravity and small perturbations of the metric and so on).  But I was wondering about: since electromagnetic radiation is composed/carried by Photons (or better:the EM force), shall gravitational waves be composed/carried by Gravitons?   In the end:do gravitational waves imply the existence of gravitons? Or it's something unrelated and off topic? Is there a description of Stack Exchange/Stack Overflow sites? Say an airplane flies at Mach 1. The doppler effect causes the front of the sound emitted by the plane to reach infinite frequency:    $f = \left ( \frac {343+0}{343-343} \right ) f_0 = ∞$  If the plane suddenly stopped, and there was an observer in front of the sound waves to receive the sound, what would the bunched up waves sound like to the observer? Would they cancel each other out and cause no sound? Get the index of the nth occurrence of a string? Windows 7 Automatic Log-on with a Default User Account? I have two User Accounts for my Windows 7 with no password requirement. I frequently use one of those accounts; so I would like Windows to log-on automatically with that account without the need for user input.  P.S. I don't prefer installing/using a third-party software for that purpose; especially those that place at low levels before Windows log-on. Is a function defined at a single point continuous? Is a function defined at a single point continuous?  For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for $f$ to be continuous, it has to have the limit $\lim\limits_{x_0\to x}f(x_0)=f(x)$ but it dosen't seem to have that limit. Let $f$ be a continuous function on $[0,1]$ such that $f([0,1])=[0,1]\times[0,1].$ Then show that $f$ is not one-one.  Hints will be appreciated. A couple years ago, it was easy to root any device: Install KingRoot, KingoRoot, or a similar app, hit the large ROOT button, and let it do the rest. After 5 minutes or so, a green "Root Succeeded!" prompt will pop up, and we can choose to keep the root app, or uninstall it and install a clean root manager app like SuperSU/Superuser.  How do they work? Why aren't they working anymore? And if they work, should I use them or not? This question arises when I try to describe my itinerary to ticket service in the UK. Take for example, I plan to buy a ticket from A to B. B is the terminal. How to say its counter part A? I cannot find a proper term. I have to say "I will depart from A" or "I plan to go to B from A". Or I even use "departure place" to refer to B. How to find the list of all available packages along with the lastest version number available in the repositories? I want to list all the packages(installed or not installed) along with the latest version number available in the repositories.  For example:  The output of apt-cache policy chromium-browser shows like below  $ apt-cache policy chromium-browser chromium-browser:   Installed: (none)   Candidate: 34.0.1847.116-0ubuntu2   Version table:      34.0.1847.116-0ubuntu2 0         500 http://ftp.cuhk.edu.hk/pub/Linux/ubuntu/ trusty/universe amd64 Packages   So the latest version of chromium-browser package available in the repository is 34.0.1847.116-0ubuntu2. Like that, i want to list the version number along with the package names that are available in repositories. And all i want to do this through command-line.  I want the output to be like this,  chromium-browser 34.0.1847.116-0ubuntu2 xxxxxxxxxxxxxx   yyyyyyyyyyy ............     ............ I'm running Fedora 25 and I need to take a few particular screen captures so I've read that shutter is a great tool for that. Sadly I am unable to use it for some reason. Whenever I try to take a screenshot this is the result: The moderator agreement is visible at the following URL:       I acknowledge and agree that as a moderator for Stack Exchange Inc.      I will abide by the then-current Terms of Service of Meta Stack   Exchange, and other moderator policies made available to me,      I acknowledge that I may have access to potentially   personally-identifying information about Meta Stack Exchange users and   that in connection with such access      I will use such information solely in accordance with the then-current   Privacy Policy of Meta Stack Exchange,      I will not disclose this information to anyone,      I will not store or copy this information and      I will only use such information in connection with performance as a   Meta Stack Exchange moderator for the benefit of Meta Stack Exchange.      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I need it for QGIS 2.99 According to two questions  and  asked on this site earlier there exists a nice relation:  $$\frac1{n!} \left(\int_{0}^t\mathrm dt \; f(t)\right)^n = \int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; f(t_1)\,f(t_2) \ldots f(t_n). $$  Unfrotunatly I am more interested in integrals of the form  $$\int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; f(t_n). $$  Can it be somehow brought to an unary integral?  All I have figured out by now is that  $$\int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm dt_2 \cdots \int_{0}^{t_{n-1}}\mathrm dt_n\; t_n^m = \frac{m!}{(m+n)!} t^{m+n} .$$ Background: I have the opportunity to pursue a Computer Science graduate degree at a well respected program (around top 20). I spoke with the professors at the university and I feel like I am a good fit with their research program. My ultimate pursuits are to complete a PhD in CS. I've done research as an undergraduate and it has only confirmed my notion that I want to pursue a research track career.  More background: I applied to jobs (in case I was rejected to all the programs I applied for) prior to hearing back from my respective graduate programs. I accepted a position for an industry job (this was to secure I wouldn't be both unemployed &amp; not in school).  Dilemma: Turns out my top graduate school is very interested in me, and I'm very interested in them. I would rather go to graduate school than work in industry forever. The industry job pays very well. I am split between A) working for a maximum of 1 year(It would only be 1 year, seriously I do not care about the money enough to work past a year) and B) going straight to graduate school.   My concerns are as follows, if I choose A):    Could I potentially defer my admissions? If I am not allowed to defer my admissions, would I have a good chance of re-applying and being accepted a year later? Would my potential advisers look down on me for deciding to work a year?   I already made my mind up it would be a 1 year gig if I decide A). I know some people say once you make money, you may not be able to readjust to the graduate salary pay. But I don't think that will apply to because I'm going to live very frugally with or without industry pay. If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples? How to solve a system of PDEs with zip condition?