How to solve this nonlinear difference equation $a_{n+1} = 2a_n + \frac{1}{a_n}$, $a_1 = 1$?
explicit formula for recurrence relation $a_{n+1}=2a_n+\frac{1}{a_n}$
In Russian we have idiom/saying "To shoot out of cannon into sparrows" (literal translation) which is used to convey an idea of applying too drastic measures to small problems. I believe there should be some native-English equivalents to this saying. Can you share if there are any? Why can't I cd to a quoted tilde ('~')? Writing my first script so I'm sure this is a basic question, but can someone please explain to me why I can:  cd ~ cd bin cd ~/bin cd 'bin'   But not  cd '~' cd '~/bin'   I need to cd to a directory path with a space in one of the directory names, so I need the quotes (it's Windows Program Files under wine). I can get around it with two cd commands, but why can't I put ~ in quotes?  If I type cd '~' (or cd "~") I get:  bash: cd: ~: No such file or directory How can I improve Ubuntu overall system performance? Iterate a list with indexes in Python In MTG, what happens when a player loses in multiplayer? In Magic: The Gathering, the rules for a player losing in a multiplayer game are quite complex, so I'd like to try to understand them by example. I would be happy to ask each of these situations as a separate question if that is preferred, but they all seem related, and I would love to have one place to look up all of these answers.  Let's say I'm playing with two friends, Bob and Chloe, in a simple free for all. Unfortunately, I lose. When I lose, suppose the following:   I controlled an  which has exiled Bob's , does he get his bear back? I have a  on the stack targeting Chloe, does Chloe take the 2 damage still? Earlier in the turn I cast  to prevent damage to one of Bob's . If it later takes 3 damage, will that damage be prevented? Something I controlled gave Bob a 1/1 soldier token. What happens to the token? Something Chloe controlled gave me a 1/1 soldier token. What happens to the token? I had a  enchanting one of Bob's army of . What happens to the bear? I had also played  on Bob's final . What happens to the last bear? Chloe had  my . What happens to the Clone? Are there any other situations specifically addressed by the rules? Suspend process without killing it So I've got a persistent program running in the background. Killing it just causes it to restart with a different PID. I'd like to suspend it (put it to sleep without actually killing it). Does kill -9 do this? If not, how should this be done? Idiomatic expression for a difficult choice This has cropped up several times in the past couple of months, and I've been struggling to find  a fitting word to describe this phenomenon. I'll describe it:  You have two choices(no, it's not Hobson's choice). Both of them are desirable, yet both of them each have a tiny part which is most undesirable, such as a tough condition, or a demanding clause. You have difficulty deciding which one to pick.  Whenever something like that comes up, I think of the expression "Can't burn a candle at both ends", but I'm not looking for that.  I'm looking for an expression that describes a difficult choice. Any suggestions?  EDIT: It's idiomatic. I'm quite sure it is. It's an expression of some sort. And no, it isn't "dilemma" Update Garmin Nüvi 255 maps Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$? I'm trying to search and replace a string in all files matched by grep: grep -n 'foo' * will give me output in the form: [filename]:[line number]:[text]  For each file returned by grep, I'd like to modify the file by replacing foo with bar. How Do I Compute the Eigenvalues of a Small Matrix? If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix?  NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See . Why will Ubuntu no longer measure file size unit as byte, megabyte, gigabyte, etc? I read on  that Ubuntu will no longer use the familiar file size units we all know by now (kB, MB, GB, TB) and switch to a different IEC standard (KiB, MiB, GiB, TiB). If this is true, I would like to know what's the reasoning behind this change, and the impact (if any) this change has, especially with multiplatform applications or applications run with Wine. Pre-declare all private/local variables? Close and flag links do nothing on mobile site This morning (it's about six AM here) I find that I can no longer submit close votes; the link just jumps to the beginning of the page.  I'm on iOS, using a third-party browser called Grazing, but a brief check with Mobile Safari shows the same problem there.   Here on Meta, where my score is much lower, I only get a "delete" link on my own question, and that works fine. (Oops :-))   Update: As per comments, this affects the "flag" link, too, which doesn't even scroll to the top of the page. Update 2: Latest comment suggests this affects Android, too. Update 3: This also affects desktop computers when viewing mobile (when you click "mobile" in the footer) Update 4:  mentions Windows 8, too; simply deleting platform-specific tags When I apply to a PhD program, may I ask a postdoc who closely supervised me to write a recommendation letter? My supervisor also has written one for me, but I believe the postdoc can comment on my research skills in a more detailed and precise way. Does a postdoc have enough authority to convince admission committee? shred multiple hard drives For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that  $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$    My initial approach is  Base Case: $(a_1^2+b_1^2) = a_1^2+b_1^2$ which is true. (Although it is trivial)  Prove the statement is true when $n=2$: We have $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2$$ (Thanks André Nicolas for pointing it out)   So if $a,b,c,d$ are integers, $ac,bd,ad,bc$ are all integers and integers are closed under addition and subtraction. Hence $(ac-bd),(ad+bc)$ are integers.   Inductive Hypothesis: $\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$ is true  Inductive Step: $$\prod_{i=1}^{n+1}(a_i^2+b_i^2) = \prod_{i=1}^{n}(a_i^2+b_i^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2)$$  Where $c$ and $d$ are integers.   But when we apply $n=2$, we have  $(c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (e^2 + f^2)$ where $e$ and $f$ are integers.  Hence, by the principle of induction, the statement we needed to prove is true. Moving linux install to a new computer I know that it can, in some circumstances, be difficult to move a Windows installation from one computer to another (physically move the hard drive), but how does that work on linux? Aren't most of the driver modules loaded at bootup? So theoretically would it be that much of a hassle?  Obviously xorg configs would change and proprietary ATI drivers and such would have to be recompiled (maybe?). Is there more to it than I'm thinking of?  Assume the 2 computers are from the same era, ie both i7s but slightly different hardware.  Update: Thanks for the answers. This is mostly for my own curiosity. I have my linux system up and running at work, but eventually I'd like to move to a computer that I can get dual video cards into so I can run more than 2 monitors. But not any time soon Proving that closed (and open) balls are convex Is there a way to add a timer to a Beamer presentation?