Use mathematical induction to prove that $\forall n\in \mathbb{N}$, $$\sum ^{n}_{i=1}i^{3}=\dfrac {n^{2}(n+1)^{2}}{4}$$  $$\begin{align*} \sum_{k=1}^{n+1} k^3 &amp;= \sum_{k=1}^{n} k^3 + (n+1)^2 \stackrel{\rm(IH)}{=} \dfrac {n^{2}(n+1)^{2}}{4} + (n+1)^2 \\ &amp;= \dfrac {n^{2}(n+1)^{2}+4(n+1)^2}{4} \end{align*}$$  Is it true? What to do next?
Need guidance on this proof by mathematical induction. I am new to this type of math and don't know how exactly to get started.   $$ 1^3  +  2^3  +  3^3  +  \ldots  +  n^3  =  \left[\frac{n(n+1)}{2}\right]^2 $$
I am trying to install Linux headers for Kali Linux on my machine and I have tried every possible solution on the internet but it always show "Unable to locate packages "  root@kali:/usr/sbin# apt-get install linux-headers-4.6.0-kali1-amd64 Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package linux-headers-4.6.0-kali1-amd64 E: Couldn't find any package by glob 'linux-headers-4.6.0-kali1-amd64' E: Couldn't find any package by regex 'linux-headers-4.6.0-kali1-amd64'   Here is my sources.list file :  # Regular Repositories deb http://http.kali.org/kali sana main non-free contrib deb http://security.kali.org/kali-security sana/updates main contrib non-free # Source repositories deb-src http://http.kali.org/kali sana main non-free contrib deb-src http://security.kali.org/kali-security sana/updates main contrib non-free   Uname -a output :   root@kali:/usr/sbin# uname -a Linux kali 4.6.0-kali1-amd64 #1 SMP Debian 4.6.4-1kali1 (2016-07-21) x86_64 GNU/Linux Recursive function, if and which statements Favicons in top-bar dropdown not SVG Change password on root user and user account If I change the password for the root user is that going to automatically change the password for the user account?   I found this link for root:    How do I change the user account password? $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent Hyperspace travel is a norm in Star Wars universe. Even small ships have that capability and without this capability galactic-scale economy isn't possible.  My educated estimates say that at any point of time at least billions of ships are in hyperspace across the galaxy. Now, talk about economic hubs like Coruscant during Clone Wars Era. The number of ships arriving here should be enormous. Millions in an hour? If one person small ship can have warping capability, then the number can exceed even billions in a peak hour. You can go shopping on a neighbor planet five times a day or you can live on a cheap planet even if you have a job on Coruscant (there are public transport cabs featured in Rebels TV show in case you aren't rich enough to have own ship).  Now, the famous quote:     "Traveling through hyperspace ain't like dusting crops, boy! Without precise calculations we could fly right through a star or bounce too close to a supernova, and that'd end your trip real quick, wouldn't it?"   That's fine. Calculations prevent you from hitting a star, supernova or planet which are already on map, but how does a ship avoid hitting another ship? Provided that number of ships arriving near a planet is that much high, the probability of two ships jumping to the same co-ordinate can't be zero. In fact, the probability should be very very high.  Is there a central body which controls the traffic by coordinating the hyperspace calculations of different ships? This seems unlikely because everyone seems to go anywhere they want while maintaining the destination privacy (there are countless examples). Or, is there really a traffic control body and those who travel privately actually gamble? Is there any other protocol if not traffic control body?  Has the canon ever addressed it? What about Legends? Does using links to other pages dilute the linking page's link juice/page rank? There are lot's of questions here about passing link juice to other pages, including .  I've read several of them over the months, but I always come away wondering the same question.  Does linking to another page, dilute or burn some of the link juice the linking page has already earned?  For example, if my current page has for example, a link juice of 10, and I link to another page, it passes a certain amount of juice (I'll use 90% as an example) of the linking page's juice, giving the page being linked to a .9 boost in it's own link juice.  However, does that mean the linking page's page's remaining juice is now less than 10?  While some of the questions would imply this to be the case, it seems counter intuitive as no one would ever want to link to any page outside of their own content for fear of losing page rank. Drawing a triangle from medians Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only?  Someone asked me this question, but I can not see it. Is it really possible?  UPDATE  Apart from the algebraic solution given by JimmyK4542, can anyone give me a direct construction?  I mean, it should sound like:     Draw a line segment sufficiently long. Cut the length of $m_1$ from it. Then $\ldots$ Commenting out multiple lines of code, specified by line numbers, using vi or vim I have learned from  that it is possible to use vi/vim to comment out a specified range of line numbers.  For example, suppose I have the following bash script:  #!/bin/bash  This is my very very great script   Now suppose that I want to comment out line numbers 6 through 8 (which contain the words very, very, and great) using the # comment character.  In vi/vim, I can simply type :6,8s/^/# to obtain the following:  #!/bin/bash  This is my #very #very #great script   which comments out lines 6 through 8.  My question is, is it possible to type a similar one liner that will remove the # comment character from lines 6 through 8 (but not any other commented lines in the file)?  Having said this, I realize that there is some debate about whether I am actually using vi or vim.  In practice, I open a file script.sh with the command vi script.sh.  Also, when I type the command which vi, I obtain /usr/bin/vi.  Nevertheless, when I simply type vi and press Enter, I obtain this:  ~                              VIM - Vi IMproved ~ ~                               version 7.2.330 ~                           by Bram Moolenaar et al. ~                 Vim is open source and freely distributable ~ ~                           Sponsor Vim development! ~                type  :help sponsor&lt;Enter&gt;    for information ~ ~                type  :q&lt;Enter&gt;               to exit ~                type  :help&lt;Enter&gt;  or  &lt;F1&gt;  for on-line help ~                type  :help version7&lt;Enter&gt;   for version info   which seems to suggest that I'm actually using vim.  I am accessing a remote Ubuntu Linux cluster using SSH from my PC.  I am not using a Ubuntu Linux GUI. Levels or First Differences, VECM or VAR for Ultimate Impulse Response Functions? Does one need to be affiliated with a university to publish papers? Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? What does each SPECIAL stat in Fallout Shelter do? How to implement dual-classing in a non-levelled RPG game? I'm working on an alternate ruleset for a tabletop pen-and-paper RPG (like D&amp;D). Recently I've been toying the idea of abolishing a level system. However, I don't want to remove the ability for a player to multi-class. Are there any games which don't have a levelling system, yet allow multi-classing? How do they do it successfully? 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This changes the integral to  $$ \begin{align} -\frac{1}{4}\int t^{-3/4}\,dt &amp;= -u^{1/4}+\mathcal{C}\\ &amp;= -\left(1+t^4\right)^{1/4}+\mathcal{C} \end{align} $$  So we arrive at the solution  $$\int\frac{1}{x^2\left(x^4+1\right)^{3/4}}\,dx = - \left(\frac{1+x^4}{x^4}\right)^{1/4}+\mathcal{C.}$$  Question: Is there any other method for solving this problem? Do I lose anything converting to DNG? I'd like to convert all my ARW (Sony) files to DNG for several reasons, but the only thing keeping me back from doing it is the fear of losing useful metadata during the conversion. I know that the image it's self is completely safe during the conversion but what about proprietary maker noteS? Will those stay too? And even if they do, are they actually useful and can be actually used by something other then Sony's own RAW converter?  P.S. I'm using Ubuntu so I'd prefer to use KipiPlugins' DNG converter because from what I understand it does everything the Adobe converter does, but if I'm wrong I can use Adobe's instead.  EDIT:  It looks like the LensID does carry over! With the ARW and the DNG files converted by Kipiplugins' built in converter, info is stored as "Exif.Sony.0x___". With Adobe's official converter, it moves the info to more logically named fields in the XMP, such as "LensID" and "Lens".  Now that I've figured it out, I think I might use Adobe's Official converter, just because I prefer how it reorganizes the Exif Data.  I'm not fully convinced though and I'd like to hear others take on all this. For some reason my iMac keeps turning off around 5 to 10 minutes into boot.  Also just recently it turns on but has shaded lines going down the screen. I have not installed any weird programs and i have not used it in around 3 months.   Any ideas on what the issue is, i don't have any other information about the issue besides whats happening when i use it.  The image i'm showing you was found on google but this looks exactly like mine does.  The model of my imac is the late 2011 model. Create an encrypted ZIP file in Python