Is there any way to prove that: $$ 1^2+2^2+3^2+\ldots +n^2=\frac{n(n+1)(2n+1)}{6}$$ but WITHOUT using mathematical induction? (I don't know, maybe through some creative graphical demonstration?)
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?
I need to set weights for some edges from a python script. I have tried looping through object.data.edges and setting bevel_weight values but I can't get it right. It only works if I have first edited some edge's bevel weight by hand from the 3d-view's transform window. But I would like to get it work from my script without any manual hacking.  Steps to reproduce:  Start Blender with default cube and switch to Scripting view.  Try to set weights for all edges:  for edge in C.object.data.edges:     edge.bevel_weight = 1.0   Verify that values changed:  for edge in C.object.data.edges:     edge.bevel_weight   Switch into edit mode in 3d-view and open up transform window by pressing n. Select edges to see that the change did not work (bevel weights are all zero). Switch back to object mode and read values again in scripting console to see that they were reset to zero:  for edge in C.object.data.edges:     edge.bevel_weight   Go back to edit mode and change some edge bevel weight straight from the transform window by hand. Go back to object mode and change values again from the scripting console:  for edge in C.object.data.edges:     edge.bevel_weight = 1.0   This time it works, bevel weights are 1.0 when you go back to edit mode and look from the transform window.   Any ideas? My final need is to apply a bevel modifier to only some set of edges and that's why I'm trying to mark them by using weights. Using vertex group for bevel modifier does not work in my situation, since I need to specify edges, not vertices, which to bevel.   I'm using Blender v2.71. How to Execute ShortCut From Command line in Windows 7 We came with a scenario where we have to use Shortcut file (.lnk), which is on the desktop, to execute an application for a headless device (i.e, without manual intervention).  Is there any way to execute it from Command prompt? How to get the CPU cycle count in x86_64 from C++? You "show" someone a picture. You "---" someone a song? In , we have a verb meaning &quot;to show&quot; corresponding to &quot;to see/to look&quot;, and we have a different verb corresponding to &quot;to hear/to listen&quot;:  inti tara stampa (you look at a picture.) ---- jiena nurilek stampa (I show you a picture) inti tisma' diska (you listen to a song.) ----- jiena nsemmagħlek diska (I 'show' you a song)  In English, do you &quot;show someone a song&quot; (sounds weird to me) or is there a verb that corresponds more directly to the Maltese insemmgħek? Use second argument for optional first argument if not provided in macro I'm trying to construct a command with one optional and one mandatory argument and I'd like the default if the optional argument is not provided to be #2.   Why does this not work?  (error: Illegal parameter number ...)  \documentclass{article}  \newcommand{\mycommand}[2][#2]{\label{#1}#2} \begin{document} \mycommand{mytext} \end{document}   And what would be a way around it?   EDIT: For clarification of my specific use case: I'm trying to define a command that passes a variable on as an option to an environment along those lines, where myenv can have options like optionone=X or optiontwo=Y so the default for the optional #1 should be optionone=#2 (not literally #2).   \newcommand{\mycommand}[2][optionone=#2]{\begin{myenv}[#1]\input{#2}\end{myenv}}   Maybe there's generally a better approach?   (The environment definition I'm using is from .) Why is Leia okay with being tracked to Yavin? In A New Hope, after Han and Luke fight off the TIE Fighters while escaping from the Death Star, these exchanges happen:             INT. DEATH STAR - CONTROL ROOM             Darth Vader strides into the control room, where Tarkin is             watching the huge view screen. A sea of stars is before him.                                   TARKIN                      Are they away?                                   VADER                      They have just made the jump into                       hyperspace.                                   TARKIN                      You're sure the homing beacon is                       secure aboard their ship? I'm taking                       an awful risk, Vader. This had better                       work.             INT. MILLENNIUM FALCON - COCKPIT             Han, removes his gloves and smiling, is at the controls of             the ship. Chewie moves into the aft section to check the             damage. Leia is seated near Han.                                   HAN                      Not a bad bit of rescuing, huh? You                       know, sometimes I even amaze myself.                                   LEIA                      That doesn't sound too hard. Besides,                       they let us go. It's the only                       explanation for the ease of our                       escape.                                   HAN                      Easy... you call that easy?                                   LEIA                      They're tracking us!                                   HAN                      Not this ship, sister.             Frustrated, Leia shakes her head.                                   LEIA                      At least the information in Artoo is                       still intact.                                   HAN                      What's so important? What's he                       carrying?                                   LEIA                      The technical readouts of that battle                       station. I only hope that when the                       data is analyzed, a weakness can be                       found. It's not over yet!   Leia doesn't even know if there is a weakness, let alone whether they'll find one. If she (correctly) realizes that they're being tracked, why doesn't she suggest that the Millennium Falcon go somewhere besides Yavin 4 instead of just frustratedly shaking her head? I know the plans , but it seems unlikely that continuing to fly to the secret Rebel base in the ship she's certain is being tracked is the only option. They are already in hyperspace when this conversation happens, but they apparently have time to talk for a little while after, so presumably they'd have time to stop and reroute. Should the weight of downvotes be increased? How to get Text to follow semicircular path, read left to right and oriented counterclockwise I'm using adobe illustrator, and I'm trying to get the text to orient from left to right(counterclockwise) and not upside down but I cannot get it to do that on the path I've created. Im pretty beginner with illustrator, any help would be greatly appreciated. thank you What would happen if two wizards made each other a Horcrux? Would they be invincible? How would you kill one of them? Is there any way to include an animated GIF directly in either PDFLaTeX or XeLaTeX? I realize the animate package can include animations in a PDF, but it doesn't support animated GIFs and you have to split them up manually into EPS or PNG files as far as I can tell. Vanishing of Taylor series coefficient How can I install an app given only its APK file? I downloaded the APK file for an app to my PC.  How can I install it on my phone? Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$  My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found  $$f(1-x)=(1-x)^4-2(1-x)^3+2(1-x)^2-x+1=x^4-2x^3+2x^2-x+1=f(x)$$ so $$I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ so $$2I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}+\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ then I can't,Thank you very much Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$ Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$ Since  and  you can get into a situation where your max HP is zero due to leveling up… what happens next?  What happens when your max HP is 0 because you leveled up with a negative Constitution modifier? Are you dead, unconscious, or what, and where do the rules say this? (The  assumes death, but that isn't established as fact anywhere yet.) Could a ship warp into Borg cube and destroy it? During the Wolf-359 incident, 39 starships were destroyed, and 11,000 people were killed or assimilated.  Why didn’t one of the smaller ships warp into that one Borg cube? Doesn’t warping into it destroy anything?  Even if it just warps through without touching, one could program the navicom to self destruct while the ship is in the middle of Borg cube. Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions? Let $\varphi: \mathbb{N} \to \mathbb{N}$ be the totient function.   Is there an integer $N &gt; 0$ such that there are infinitely many integers $n &gt; 0$ such that $$\varphi(n) = N?$$ How "submitted", "to appear", "accepted" papers are evaluated in a CV? Assume a search committee is reading my CV and in the publication section they notice some of my papers are only submitted or claimed to appear in a journal (or accepted for publication in a journal). Sometimes the journal which has accepted the paper for publication lists the title of accepted articles before actually publishing them, but in the rest of cases there is no evidence to prove that the referee process of the paper is over and the journal has accepted the paper for publication. Also assume preprints of my papers are available in ArXiv. So, my questions are:   How a search committee interprets and evaluates those papers which are just submitted or are claimed to appear?  Does a search committee consider these types of publications less valuable than the published ones (assuming the same quality)?  Does a search committee refer to my preprints in ArXiv to evaluate my submitted or accepted papers? Can I open Microsoft Word and Excel files? Can I open Microsoft Word and Excel files with Ubuntu?