Blocks of Pyramid Pattern Expression There is a pattern following, and trying to find the algebraic expression Each layer (from the top). Diagram.  So the first layer has 1, second has 4, third has 9, and the fourth has 16. That's how the sequence is increasing. What I'm looking for is, When the second layer is added with the first layer, Third layer is added with the second and first, Fourth is added with third,second and first. So something like this.   I am trying to find the algebraic expression for this pattern. Any ideas?? Thank you
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ 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?
Does xbox live gold membership stack when I purchase a new one? I currently have an Xbox Live Gold membership that will expire in March 2011. However the price of membership is increasing to $59.99 starting November 1, 2010.   If I buy a 12 month membership now at the current price of $49.99, will the additional 12 months be added on to my existing membership or will I start a new membership which will expire 12 months after I buy it? How to modify itemize vertical spacing within blocks in beamer? I have noticed that when I have "itemize" environments defined within blocks in beamer, that they introduce more vertical spacing between the top and the bottom of the blocks, than vertical text. Since I have a slide with some text filled blocks and directly after, a slide with only "itemize" holding blocks, I would like the spacing between the itemized text and the block to be the same as for regular text.   On  I have read about "squeeze" option, but it doesn't seem to work...   Additional info:  The problem is in the additional vertical space that is included by the "\itemize" environment. I have found  on how to remove vertical spacing from the itemize list, and I am using this approach. However, it is a brute force approach. What I would like to know if it is possible to have the same vertical space between the end of the itemize list and the bottom of a block, as the one set between the normal text and the bottom of the block.   MWE:   \documentclass[10pt]{beamer} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{exscale} \usepackage{epsfig} \usepackage{fancybox} \usepackage{lmodern} \usepackage[T1]{fontenc} \usepackage{overpic}  \usepackage{multimedia}  \usetheme{default} \usecolortheme{crane}  % -- Modifications that enable the footline and set black color for the fonts  \setbeamertemplate{footline}{%     \begin{beamercolorbox}[wd=\paperwidth,ht=2.25ex,dp=1ex,right, rightskip=1mm, leftskip=1mm]{titlelike}         \textcolor{black}{             %\inserttitle\hfill\insertauthor\hfill\insertframenumber%             Something\hfill Something Else \hfill\insertframenumber%         }     \end{beamercolorbox} }  \setbeamertemplate{itemize items}[square]  \setbeamercolor{item}{fg=black} \setbeamercolor{frametitle}{fg=black} \setbeamercolor{block title}{fg=black} \setbeamercolor{title}{fg=black} \setbeamercolor{block body}{bg=white}  % END Modifications  \begin{document}   \begin{frame}   \frametitle{text blocks}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   blockk     \end{block}{block}      \begin{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block} \end{frame}  \begin{frame}   \frametitle{itemize blocks}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}   \end{frame}  \end{document} Splitting a string into words and punctuation I'm trying to split a string up into words and punctuation, adding the punctuation to the list produced by the split.  For instance:  &gt;&gt;&gt; c = "help, me" &gt;&gt;&gt; print c.split() ['help,', 'me']   What I really want the list to look like is:  ['help', ',', 'me']   So, I want the string split at whitespace with the punctuation split from the words.  I've tried to parse the string first and then run the split:  &gt;&gt;&gt; for character in c: ...     if character in ".,;!?": ...             outputCharacter = " %s" % character ...     else: ...             outputCharacter = character ...     separatedPunctuation += outputCharacter &gt;&gt;&gt; print separatedPunctuation help , me &gt;&gt;&gt; print separatedPunctuation.split() ['help', ',', 'me']   This produces the result I want, but is painfully slow on large files.  Is there a way to do this more efficiently? What is the origin of "that's using your noggin"? Using File Geodatabase coded domain with ArcGIS Online? I created a File GDB and set up a domain, and the domain was assigned to a field in a feature class. The domain behaves as expected on ArcGIS Desktop (dropdown list when editing the content of that field).  But if I zip up the FGDB and upload to ArcGIS Online to created a feature layer and edit the layer online, there is no dropdown when editing.  I don't see any additional step in this tutorial, other than a different mechanism of getting data to ArcGIS Online. I don't think I can do it the same way in ArcGIS 10.0:   Anyone has any idea what's going on? Print file content without the first and last lines Is there a simple way I can echo a file, skipping the first and last lines? I was looking at piping from head into tail, but for those it seems like I would have to know the total lines from the outset. I was also looking at split, but I don't see a way to do it with that either. Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls. I know that every open set in $\mathbb{R}$ can be decomposed as the union of countably many disjoint open intervals and I'm trying to show that this property no longer holds in higher dimensions. For this I'm trying to show that there is no countable, pairwise disjoint open balls whose union is the open square $(0, 1)^2 ⊂ \mathbb{R}^2$. I know that it is because of the corners, but I don't know how to write this in mathematical terms. Thanks for any help. "is" vs "are" when followed by a number I read the following sentence on YouTube and initially thought it was grammatically incorrect:     Here are 10 minutes of the movie Black Hawk Down!   I thought "are" should have been "is" but then I realized minutes is plural so the plural form, are, is used correctly. I also thought how most people would naturally say "Here is ten dollars in change" but is the use of the singular form, is, correct since dollars is plural?   Am I misunderstanding a rule? Where can I get a clean ISO of a specific build of Windows 10? Command prompt as SYSTEM user (Windows 7) How can I get a command prompt as the SYSTEM user on Windows&nbsp;7?  Context: I'm testing a computer startup script, and I need to figure out the behavior of a command when run as SYSTEM rather than my user account.  I've seen ways to do this with at and psexec, but those methods only appear to work on Windows&nbsp;XP or Windows&nbsp;Vista.  I'm using Windows&nbsp;7 Pro SP1. Fubini's theorem for series with dependent indices As a part of a proposition about power series that I am trying to prove I have to show that  $$ \sum_{m=0}^{\infty} \left(\sum_{n=m}^{\infty} \frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n\right)(x-b)^m = \sum_{n=0}^{\infty}c_n(x-a)^n$$  Proof: What I have done so far is to show that  $$\sum_{m=0}^{n} \frac{n!}{m!(n-m)!}(b-a)^{n-m}(x-b)^m = (x-a)^n $$  So the next idea is to use Fubini's rearrangement theorem for infinite series (I have also shown that the series are absolutely convergent). However, the theorem does not accomodate for dependent indices. Intuitively (using triangular summation), I think that the proper way to interchange series would be \begin{equation*} %\begin{array}{ll} \sum_{m=0}^{\infty} \sum_{n=m}^{\infty} \frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n(x-b)^m = \sum_{n=0}^{\infty} \sum_{m=0}^{n}\frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n(x-b)^m %\end{array} \end{equation*} and thus  $$=\sum_{n=0}^{\infty} c_n(x-a)^n$$ However, I do not know how to perform this step rigorously in case of infinite summation. How come SG-1 couldn't get back to Earth? In Stargate SG-1 S02E16, , SG-1 (Captain Carter, Teal'c and two other members, who replaced Colonel O'Neill and Dr Jackson) is "trapped" on planet in binary system, where second star has risen and is slowly raising temperature - Samantha Carter says, in one moment, that it's going to reach 200°F soon - and dialing device is locked mid-dial. Yet, SGC manages to dial the planet and send them instructions how to repair the device and off-world team makes it back.  Why didn't they come back when Earth first dialed P9Q-281 to contact SG-1? I mean, from my understanding, stargate wormhole is two-directional, meaning it doesn't matter which end it was activated from, yet it seems like it is one-directional from the events of this episode... how to disconnect main service drop at service panel when upgrading to a 200 amp service panel with service drop coming from bottom underground do I need southern California Edison to come disconnect and then reconnect or can I just do it on my own. I do have a permit. Is it possible to find a smooth function $p$ such that $$p(x)=0 \text{ if } x\le -1$$ $$p(x)=0 \text{ if } x&gt; 0$$ $$0\le p(x) \le 1 \text{ if } -1&lt;x&lt;1$$ $$p(x) = 1 \text{ if } -\frac{1}{2}\le x \le \frac{1}{2}$$ Hint: The following function has continuous derivatives of all orders $$f(x) = 0 \text{ if } x\le 0$$ $$f(x) = e^{-\frac{1}{x^2}} \text{ if } x&gt;0$$  My thought: Considering the left-half of the function $p$, I tried something like $e^{-\frac{1}{\tan^2(x\pi)}}$. The shape looks good but I am not sure about the smoothness at $-1$ and $-\frac{1}{2}$. For $x=-1$, since $\tan$ is infinitely differentiable so we have smoothness there. But for $x=-\frac{1}{2}$, I am not sure about the smoothness after &quot;gluing&quot; with $y=1$. I'd never heard of anycast until a few seconds ago when I read "".  The wikipedia "" article on it is quite formal and doesn't really evoke a mental picture of how it would be used.  Can someone explain in a few informal sentences what "anycast" is, how you configure it (just in a general sense), and what its benefits are (what does it make easier)? Cannot log in with Apple ID after performing password reset My wife forgot the password to her Apple ID account. Her Macbook Air (OS X Yosemite) restarted to install some updates. She has her user account set up through her Apple ID account.  Since she forgot the password she reset it from my laptop.  She can log in to iCloud and Apple ID with the new password, but cannot log in with the new password on the Air.  When she tries to log in she is greeted with this message:     "This account uses an iCloud password to log in. If you need to change   the password, use another device to change it at appleid.apple.com." Repeat the same reference in footnote on different pages I was listening to the song  by Eminem. I've listened it before many times but from the time I've started learning English grammar I am getting confused while reading or listening English. This time I got stuck at the line "Too much pride, between you and I Not a jealous man but females lie". The phrase "between you and I" always sounded correct to me. But recently I learned that me is only used as an object and hearing that phrase sparked my head. Then I searched over the net and found on  that it is actually a hypercorrection.  says,      "...according to many grammarians and stylists a pronoun in a prepositional phrase in English is supposed to be in the oblique case...   ...though there is still disagreement on whether the phrase itself in today's language is grammatically correct or not."   My first question is, why is there a disagreement? Don't we always use the object form in the oblique case? If not then what are those cases?  Secondly, I want to know how common the phrase "between you an I" is? Is it used colloquially by common native English speakers. By common speakers I mean who don't bother grammar. I searched on  and got some results, e.g. . I found the phrase in another song -- . Is it a common mistake among native English speakers or is it used to give some special linguistic affect? I think this phrase has some linguistic affect that is why the singer would have given the name of her song "Between You and I". Because no one would like to use incorrect English in the title of a song, it must have some purpose. Image UriSource and Data Binding How do you use script variables in psql? In MS SQL Server, I create my scripts to use customizable variables:  DECLARE @somevariable int   SELECT @somevariable = -1  INSERT INTO foo VALUES ( @somevariable )   I'll then change the value of @somevariable at runtime, depending on the value that I want in the particular situation. Since it's at the top of the script it's easy to see and remember.  How do I do the same with the PostgreSQL client psql?