Task from old exam:     Prove that $$\sum_{k=1}^{n}k! \cdot k= (n+1)! -1$$   I'd like to use induction proof and nothing else.  Here is what I did:  $n=1:$ $\sum_{k=1}^{1}1! \cdot 1= (1+1)! -1 \Leftrightarrow 1=1$  So this is true, so it will work for an $n$ already.  It worked for one $n$, now we need to show it will work for every other $n$, too.  $$\sum_{k=1}^{n+1}k! \cdot k= (n+2)! -1$$  But what can I do from this point on? There is this sum symbol on the left and I don't know how to continue with it. Any hints maybe?  I really don't want know another way of solving it. It might be easier with telescope or something else but the recommendation was induction.
Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated.
I've gone into the Nether and seemed to have lost my portal out of it, I also can't die in lava or by mobs attacking me, I'm appear to be invincible and can't die.  This is Single Player by the way.  Why is this happening? Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $. So we obviously we want $ x^{2} + y^{2} + 1 \equiv 0 ~ (\text{mod} ~ p) $.  I haven’t learned much about quadratic congruences, so I don’t really know how to go forward. I suppose you can write it as $ x^{2} \equiv -1 - y^{2} ~ (\text{mod} ~ p) $, but again, I’m not sure where to go. I know about the Legendre symbol and quadratic residues, which I know are involved in this question, but I don’t know how to apply what I know. :( Firefox application associations not working in the 'Downloads' window No matter what changes i make to file associations (actions) in the 'Applications' tab in Firefox, they're totally ignored. For example, i set .wmv and .avi files to open with 'smplayer' but when i download a file and double-click on it (through the 'Downloads' window), it keeps opening with Totem player.    The default file associations through nautilus ('open with' tab) work fine, i.e. i've set smplayer as the default player and when i double-click a video file it open correctly.  I've tried to delete and recreate mimetypes.rdf but that didn't help.  Any ideas on what else should i check? Get mp3 duration in android The impression that I got from the  films was that if  was successful, most of the world was going to suffer greatly. Basically everyone would serve Voldemort as slaves or die. So what's the point of serving someone who's probably going to ultimately end up betraying and/or enslaving you? I know the  hated , but is that really enough to justify tearing down Wizard society? Why exactly do the Death Eaters serve Lord Voldemort? Fear? Is the post-Voldemort world not that bad? (I for one welcome our ) Using adjustbox along with table footnote Nested amsmath environments with \newenvironment How do I install a .tar.gz (or .tar.bz2) file? Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact.  This has been asked before, but all the answers I have seen prove the contrapositive. Realistically, this may be the way to go, but is there way to exhibit an unbounded continuous function (under the assumption $X$ is not compact) without appealing to results beyond introductory real analysis (e.g., the solution I've seen involves the Tietze Extension theorem)?  Because we're working with metric spaces, it's clear that the assumption that $X$ is noncompact (towards proving contrapositive) will lead us to extract a sequence of points in the space with no convergent subsequences. But how can we infer the existence of an unbounded continuous on the space knowing only about some sequence of points in this space (without using anything too advanced)? Lost email settings and no icon for Mail I've lost my Settings → Mail, Contacts, and Calendars on my iPhone 5.  No icon for Mail and no Mail settings.  How do I get it back? See a list of my top voted comments? Or another way to track progress on Pundit? Pundit  Left 10 comments with score of 5 or more   I saw a discussion question that mentioned Pundit, which that stated that , however, something like top voted comments seems to fit in with the rest of stackoverflows feature set, as well as indirectly provide a method to track progress on this badge. Should there be (or is there already) a list of my top voted comments somewhere? I'm trying to connect to another computer in my house via ssh. Both computers have ubuntu and openssh installed. When I run the command "ssh username@ip" on terminal I'm asked for the user's password I'm trying to connect to. I type it but I get this error "Permission denied, please try again." I'm sure the password is correct, then why can't I access? I've read many years ago in books, that the brain has no nerves on it, and if someone was touching your brain, you couldn't feel a thing.  Just two days before now, I had a very bad migraine, due to a cold. It's become better now, but when I had it I felt my head was going to literally split in half, as the pain was literally coming from my brain.  So it lead me to the question: How come people can get headaches if the brain has no nerves? What made the Ministry admit Sirius was innocent? I have a problem analysing this sentence from the point of finite/nonfinite clauses, clause elements and their functions:     He does not want to destroy his parents' dream of him achieving a Cambridge degree.   I am especially interested in the: dream of him achieving a Cambridge degree.  I know that 'achieving a Cambridge degree' is a non-finite -ing participle clause. However what is its function? And what is the function of 'of him'? Is it a postmodification? If every element of set A ia an element of set B then we say that A is subset of B. A is said to be proper subset if A does not equal B.  My question is - whether the symbol we use to show subset is  same as that for proper subset? I have seen that A is subset of A itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this. Can I ignore coefficients for non-significant levels of factors in a linear model? In setting up dual Boot with Windows and Ubuntu, which OS do I install first? I'd like to install both Ubuntu 12.04 and Windows XP on a Dell laptop, and I was thinking about using a dual boot structure, and using the bulk of my hard drive as empty hard drive space to share files between the two operating systems (so choice of file system type is very important in this set-up).     The kind of partitioning structure I would like to use is  Partition 1 - Ubuntu 12.04 (root) (20GB)  Partition 2 - Ubuntu /home        (20GB)  Partition 3 : Free Space          (560GB)  Partition 4 : Windows XP          (35GB)  Partition 5 : SWAP                (3GB)  (Total Hardrive Capacity is ~640GB)    My question is; what is the best way to go about setting up this kind this system?   Should I install Windows XP first and setup the partitions, and then install Ubuntu which I believe will install the GRUB bootloader for OS booting choice   or   Do I install Ubuntu first, setting up the available partitions and then perform a WIndows install?  Please let me know if there is anything in this setup that I have left out and should know about, including things related to setting particular partitions as logical or primary, and whether the boot partition and the filesystem partition should actually be two separate partitions. Are there systems in classical mechanics that are provably non-deterministic? Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.