I need to prove $$\sum_{k=1}^n(2k)^2 = \frac{2n(n+1)(2n+1)}{3}$$ using mathematical induction. I tried solving this but I got stuck when proving that this is true for n=k+1.  I would be very thankful if someone could help me.
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?
How to run a complete GUI as root? LuaLaTeX protrusion problem with Negative value Pathologies in module theory Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. Can I travel to Germany from a country different from the one I got the visa from? For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$? The motivation for this question is in my answer to this post :       (1) If I take $A$, $B$ and $C$ three groups such that $A\times B$ is isomorphic to $A\times C$ then do we have $B$ is isomorphic to $C$ ?   In the post a counter example is given when $A$ is an infinitely generated group. However I cannot get a counter-example when $A$, $B$ and $C$ are assumed to be finite groups. Note that there is no reason a priori if $\phi$ is an isomorphism from $A\times B$ to $A\times C$ that $\phi(A\times\{1_B\})=A\times\{1_C\}$.   In the case of finite groups it is true because of the Krull-Schmidt theorem : .  A slightly more general question is the following :     (2) If $B$ and $C$ are two finitely generated groups, can I always find a finitely generated group $A$ such that $A\times B$ is isomorphic to $A\times C$?   If $A$ is not assumed to verify any property but to be a group we can take :  $$A=B^{\mathbb{N}}\times C^{\mathbb{N}} $$  Furthermore replacing :$ \prod_{\mathbb{N}}$ by $\bigoplus_{\mathbb{N}}$ (or more likely the restricted product for groups but I am not sure of a well known notation, it is the subgroup of the product with all but a finite number components being trivial) will give you the same result for countably infinite groups.   $\underline{\text{Edit}}$ : To be clear about what I am asking. (1) is kown to be true for finite groups whereas (2) is known to be true when we are considering groups without restriction (even just for countably infinite groups). Let us pretend we are working on the smaller class of groups : the finitely generated groups. Is there any counter example to $(1)$? Is there any counter example to $(2)$?   $\underline{\text{Edit 2}}$: Here is a partial answer, from the classification of finitely generated abelian groups $(1)$ is always true if all groups are assumed to be abelian finitely generated groups. Now assume that $A\times B=G=A\times C$. Then you have (derived groups) :  $$D(G)=D(A)\times D(B)=D(A)\times D(C) $$  So the abelianizations verify :  $$A^{ab}\times B^{ab}=A^{ab}\times C^{ab}$$  Because we are now in the realm of finitely generated abelian groups we get :  $$B^{ab}=C^{ab} $$  So $(2)$ cannot be true in general for instance you can take $B:=\mathbb{Z}$ and $C=\mathbb{F}_5$, those two groups have different abelianizations.   The new question can be found here : Is it possible to copy text from image? Is there any tools to copy text from video's or images? I have several thousand nodes of a particular content type. Using the web interface (example.com/admin/content), I can only delete about 50 at a time. How can I quickly delete them? Performing deep learning land cover classification using R? I am really new to Deep Learning and, unfortunately, I can't find example codes on land cover classification other than  where the author wrote a script in R for a large dataset.   The main reason that I am asking is because recently I found a few papers on Remote Sensing Image classification using Deep Learning and I was wondering if there were any R examples on that subject.  So, is there a code example, preferably a step-by-step, which I can use?  I found several git repos like  that can do Deep Learning classification with R, BUTT, I am still looking for a simple multi-spectral image segmentation method. An example of that can be found at  and I also found a similar question in  for which the answer is "use something else other than R if you can".  If I find anything that works today I will be posting a working answer. I have a SRD-12VDC-SL-C relay with this configuration:  And I want to trigger it with PIR Motion Sensor Detector Module HC-SR501  The problem with direct connection to PIR is, when I connect PIR with 12v DC the Relay remains always ON, i.e. trigger from PIR does not switch it off. And when I connect it via 5V supply the Relay remains always OFF. The trigger output of PIR is 3.3V how can I rectify this problem? by adding some resistor/FETs, or something else. Edit: The voltage at Relay-In(Green) is 10.5V and the relay is OFF it need to go below 9V in order to switch the Relay ON. When I connect it to PIR trigger it goes down to 2.7V (Thus Switching ON the relay, below 9V). And when the PIR trigger of 3.3V comes it goes up to 7.2V but because it is still below 9V) Relay does not switch. sudo error, is mode 0777, should be 0440 How to horizontally center an element Agreeing on a single shortened nomenclature? I've noticed across various stack exchange sites that when they refer to their own stack exchange on their site, they'll shorten the name to something similar to either:  EE.SE (Electrical Engineering Stack Exchange) OR Tex.Sx (LaTeX Stack Exchange)  Essentially they'll do XX.SE or XX.SX which has the possibility to generate some confusion for new users.    Have I not visited enough sites and just noticed an outlier or would it be beneficial for Stack Exchange sites to agree on a shortened name nomenclature? Extreme of $\cos A\cos B\cos C$ in a triangle without calculus. If $A,B,C$ are angles of a triangle, find the extreme value of $\cos A\cos B\cos C$.   I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing  helps.  On this topic we learned also about Cauchy inequality, but I have no experience with it.  The answer according to Mathematica is when $A=B=C=60$.  Any ideas? I've installed the GNOME Shell in Ubuntu 12.04 and I want to change the "Shell Theme", But it doesn't allow me to do that.  The alert says something like: "The GNOME-Shell extension is not activated", and I want to install it with the console, But it returns me an error.  How can I solve this problem? Why is spectrum obtained by sunlight, said to be continuous? My teacher spoke about atomic spectra today, and he explained that, unlike the spectrum obtained by analyzing the sunlight, the spectra of atoms are not continuous.     I have a question about this - even the sunlight has the radiations emitted by the atoms of the elements which compose the sun, still spectra is continuous, which is in opposition to the statement that atomic spectra is discontinuous. So, spectrum obtained by sunlight is continuous even though it atomic spectra. In order to account for spectrum obtained by the sunlight to be continuous and atomic spectra to be discontinuous, can we confirm that sun consists of all those elements (sodium, helium, neon, mercury, etc) which emit the colors of frequency belonging to visible region? The situation is my  has been enchanted with .  The creature has      when it dies, return it to his owner’s hand   The enchantment says     When enchanted creature dies, return that card to the battlefield under your control with a +1/+1 counter on it.   Who gets the creature? Me or my rival?  Does the creature ability trigger before or after enchantment? Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for- SQL to find the number of distinct values in a column I can select all the distinct values in a column in the following ways:   SELECT DISTINCT column_name FROM table_name; SELECT column_name FROM table_name GROUP BY column_name;   But how do I get the row count from that query? Is a subquery required? I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. What should I do if I have two libraries that provide functions with equivalent names?