I am trying to prove that $\sum_{k=1}^n (2k)^2 = \frac{2n(n+1)(2n+1)}{3}$. I managed to do the base case but I am stuck at the inductive step. Am I doing it wrong? This is what I need to prove: $$2^2 + 4^2 + \cdots + (2n)^2 = \frac{2n(n+1)(2n+1)}{3}$$ Here is the working I did. I feel that I did something wrong but I am not sure which portion. Show $k+1$ is true: $$2^2 + 4^2 + \cdots + (2k)^2 + (2(k+1))^2 = \frac{2(k+1)((k+1)+1)(2(k+1)+1)}{3}$$ $$\frac{2k(k+1)(2k+1)}{3} + \frac{3(2(k+1))^2}{3} = \frac{2(k+1)((k+1)+1)(2(k+1)+1)}{3}$$
I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof?
Single word to denote date and time I want one word that means Date and Time. For example in "the date and time of the event will be announced later" I want to use single word instead of "date and time" Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$    I think this is known (see ), I appreciate any hint or link for the solution (or the full solution). How do I check if a list is empty? How to preserve bookmarks when rearranging pages of a PDF file with tools like pdftk? I am using pdftk to rearrange pages of a pdf file with bookmarks/outlines, but after that, the output file lost the bookmarks of the original.   My command is pdftk in.pdf cat 1-22 43 23-42 44-end output out.pdf.  I was wondering how I could keep the bookmarks while rearranging pages? I always see commands like this  apt-get -y install    What is the -y tag for and what is the difference between apt-get install with and without the -y tag? How to fix Terminal error DYLD_ environment variables being ignored because main executable (/usr/bin/login) is setuid or setgid? What is the non-parametric equivalent of a two-way ANOVA that can include interactions? Hi I am trying to find the non-parametric equivalent of a two-way ANOVA (3x4 design) which is capable of including interactions. From my reading in Zar 1984 "Biostatistical analysis" this is possible using a method put forth in Scheirer, Ray, and Hare (1976), however, according to other posts online it was inferred that this method is no longer appropriate (if it ever was).  Does anyone know what method would be appropriate for doing so, and if so the corresponding functions in R or Stata? Smoke Generator does not work When I try to run the smoke simulator in Blender 2.62 I get no smoke at all. When I attempt to render, the only output is a black screen. What am I doing wrong here that is making the smoke simulation not render?  Is it a mistake I made, or a problem with Blender version 2.62? How to ensure that all traffic goes through VPN connection once it has been started? I connect to the internet through a VPN provider along with other precautions in order to ensure privacy (against traffic analysis by agencies of a repressive government). I currently use  to configure and establish the VPN connection. Tunnelblick does not allow automatically connecting to the provider at or before login so I need to connect manually each time, and sometimes the connection is lost.  I need any and all traffic to go through the VPN connection at all times; whenever the VPN isn't connectable for whatever reason, I want the fallback to be no connection. I need to ensure   that OS X does not connect to the internet on my regular internet connection, or at all, during boot (until the login screen, for time synchronization and other possible "call home" stuff etc.). What services, if any, try to do this, and how can I practically analyze and change what's going on with networking during boot? Is there a way to connect to VPN earlier than right after login? that if my VPN connection is temporarily lost, applications do not continue to communicate over my regular unencrypted connection.    I basically want the system to act as if the network connection is lost altogether if the VPN connection fails, but currently they just continue to work as if nothing happened. What can I do to my system so that once a VPN is established, all traffic can only flow through that connection and no traffic can flow if the connection drops? c stdout print without new line? I was just looking at the statistics for migrated questions in the  mod tools, and noticed this oddity:    The actual statistics, as visible to any 10k+ SFF user , are: 6 migrated to Worldbuilding, 2 to Literature, 2 to SFF meta, 1 to Movies &amp; TV, 1 to Writers, and 1 to Aviation.  Why is one of the meta migrations showing up with a blank icon rather than with the other?  Both migrated questions were closed as duplicates on meta. Both were migrated by 5 community members without mod involvement. One of the stubs on main has since been deleted, but so have the stubs for some of the other migrated questions. I can't think of anything to explain this discrepancy. I'm sure there must be a simple reason for this but I can't seem to be able to import a shp file or a feature class from another file or sde geodatabase into a Feature Dataset without getting an error.  The feature classes come in fine if I don't don't try to import them into a Feature Dataset.  The feature dataset definitely has the same projection as the feature classes I'm trying to import so I don't think that's the problem.  The Error I get is ERROR 000210 and just says Failed to convert and cannot create output and Failed to execute (CopyFeatures).    Any ideas what I may be doing wrong here.  Seems like something so simple.  I'm trying to use the Feature Datasets to better organize my data which I think is there purpose.   Update:  So as you can see below, Jason helped me find the answer.   As suggested by Matt below, I added a new Idea to the ArcGIS Ideas site.  Here is the link to that, so please promote this up if you agree a better error message is needed here: Is it possible to use a spell scroll with a spell that requires a bonus action to cast? The DMG contains spell scrolls (on page 200), with the following rule:     If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components.   This is all well and good for spells with a casting time of 1 action (which admittedly is most of them), but for spells with a bonus action casting time, how is it supposed to work?  Does it require:   Both an action and a bonus action?  Just an action? Just a bonus action? Something else I haven't thought of?   In case it needs saying, I'm not interested in speculation here - only RAW-based answers or explicit statements of developer intent should apply. How can I change the default "ens33" network device to old "eth0" on Fedora 19? Why did the gamma ray burst from GW170817 lag two seconds behind the gravitational wave? , reporting on the announcement of gravitational wave GW170817, explained that for the first time we could identify the precise source of a gravitational wave because we also observed the event in the electromagnetic spectrum. It notes however that the gamma ray burst detected by the FERMI space telescope was observed nearly two seconds later than the gravitational wave.   Did the gamma ray burst actually arrive at Earth two seconds after the gravitational wave, or is this time delay just some kind of observational artefact? If the delay is real, what is its cause? Is the delay due to the gamma ray burst somehow being slowed, or was it 'emitted' at a different time in the merger event? What does a delay even mean when the gravitational wave was detected for over 100 seconds? Is it a delay from peak GW to peak gamma ray? Are answers that just contain links elsewhere really "good answers"? I have often wondered at the practice on Stack&nbsp;Overflow, Server&nbsp;Fault, and the other sites where someone will ask a question and an answer will come in that says Look here: http://link elsewhere. Rarely will the link have exactly the answer necessary -- often it's an entire article -- yet these answers get voted up.  Why is that, exactly? I realize this is subjective, but my own thoughts are that one should try and give the answer (even if the answer is 'no, it can't be done' or 'no, I don't think it can be done' -- very different answers) and then if you have a link that explains it, go ahead and post it within the answer.  What does the community think about this?     For more information, see  in the . Fastest way to determine if an integer's square root is an integer I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because . What language is the Ubuntu Software Center written in? Just been wondering, since elementary AppCenter is written with Vala and Gtk+3.  Also, what GUI wrapper does it use? Related questions with over 99 score display incorrectly The score for the 4 questions are 111, 1218, 261 and 5 respectively, but they display 2 digits per line. This is an obvious bug in design. GTA V Online Character Transfer PS3 to PS4 I recently just bought a PS4 that I couldn't pass up (Price Wise) and decided to try and transfer my GTA V Online Character.  Stats: All stat bars full $204,000,000 Rank: 149  I've been playing the game since it came out, which was around 2013 I believe. I quit playing the game about 8 months ago because my hard drive in my PS3 died on me so I decided to put the console on the shelf for a while. Like stated above, I just bought me a new PS4 and was looking to try and transfer my character over. I tried calling Rockstar Support about my situation and all they told me was that they don't allow transfers anymore.   So here's my question... Is there absolutely any way that I can transfer my character to my next gen console?