Find all function $f: \Bbb{N}_0  \rightarrow \Bbb{N}_0 $ satisfying the equation $f(f(n) + f(n) = 2n +3k$ for all $n \in \Bbb{N}_0 $ where $k$ is a fixed natural number.   I have proceeded with solution to a point but I am not able to go further  so lets say $f(0)=m$ $\implies$ $f(m) =3k-m$ $f(3k-m)=f(f(m)=2m+3k-f(m)=3m$ $f(3m)=f(f(3k-m))=2(3k-m)+3k-f(3k-m)=9k-5m $  after this I am not able to proceed through
Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all &amp; $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number.  A friend of mine suggested me to use recurrence relation. While it is unfortunate that I have problems understanding recurrence relation. Please refer information to understand difference equation, homogeneous equation &amp; auxillary equation.  Thanks in advance! Kindly don't down vote for not showing my attempt. I am seriously not able to understand the question and the process suggested to use.
Ever since I upgraded to 13.10, the software updater does not show all the available updates. Some will appear and can be installed just fine, but once it claims that all software is up to date, using sudo apt-get upgrade shows that there are still more packages waiting to be updated. I can update them via the command line, but that only solves things temporarily. I'm not sure why some show up and others don't. How can I make Update Manager update all packages?  Update-manager shows this:    But apt-get shows this:   Thanks! I saw there are various manga with Akame ga KILL! in their names.  The ones I have found are:       Which one should I read if I want to continue the storyline of the anime? Why prefer the 18-55mm and 55-250mm lenses vs 18-200mm? If light is an electric and (magnetic field), how can it be absorbed? I was wondering how light or any electromagnetic radiation can be "absorbed" if it consists of electric and magnetic fields. For example if there is a charge at point A, and the  light reaches point A, it will cause that charge to move depending on the value of the electric field. However, why wouldn't this electric field continue on and influence others particles everywhere else? For example, if you have a static charge creating an electric field, then if you put a charge at point A it will feel its effects and then if you bring a charge at another location B it will also feel its effects, namely it will not have been absorbed. Please shed some light on this. Thanks  edit: Does this mean the electron that absorbs this light moves in such a way that it creates an electric field of its own canceling the initial one? Or am I way off? Since Chrome version 38, there's a pretty ugly button in the title bar of the Chrome window that allows you to switch profiles quickly.  Since I'm the only one that uses my computer, this is not really useful to me, and I'd rather reclaim the space for more open tabs.  I've looked in the settings but don't see a way to disable it.  How can I get rid of the button? How to make child collide ufw: how to forward single port to another port on different interface I have two interfaces eth0 (192.168.1.2) and eth1 (192.168.3.2).  I receive data from 8080 at eth0. I want to forward it to eth1 port 80.  I'm using UFW, because i don't understand iptables and how to add it at the beggining of /etc/ufw/before.rules.  I've already edited DEFAULT_FORWARD_POLICY="ACCEPT" on /etc/default/ufw and already edited net.ipv4.ip_forward=1 on /etc/ufw/sysctl.conf How to turn a ruled pattern into a multivariable function How does an internet server respond to a request from a private IP? Ruby can add methods to the Number class and other core types to get effects like this:  1.should_equal(1)   But it seems like Python cannot do this. Is this true? And if so, why? Does it have something to do with the fact that type can't be modified?  Update: Rather than talking about different definitions of monkey patching, I would like to just focus on the example above. I have already concluded that it cannot be done as a few of you have answered. But I would like a more detailed explanation of why it cannot be done, and maybe what feature, if available in Python, would allow this.  To answer some of you: The reason I might want to do this is simply aesthetics/readability.    item.price.should_equal(19.99)   This reads more like English and clearly indicates which is the tested value and which is the expected value, as supposed to:  should_equal(item.price, 19.99)   This concept is what  and some other Ruby frameworks are based on. Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$ Let $\{a_n\}, {n\geq 1}$, be a sequence of real numbers satisfying $|a_n|\leq 1$ for all $n$. Define  $$A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n),$$ for $n\geq 1$. Then find $\displaystyle\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ .    I proceed in this way  $$\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)=\lim_{n \rightarrow \infty}\sqrt{n}\left[\frac{1}{n+1}(a_1 + a_2 + \cdots + a_n+a_{n+1})-\frac{1}{n}(a_1 + a_2 + \cdots + a_{n})\right]=\lim_{n \rightarrow \infty}\left[{(na_{n+1}-a_1 - a_2 - \cdots - a_n})\frac{1}{\sqrt{n}(n+1)}\right]$$  Please help me to complete from here What does "isomorphic" mean in linear algebra? A linear operator commuting with all such operators is a scalar multiple of the identity. Associativity of symmetric difference of sets True random generation in Java I was reading the Math.random() javadoc and saw that random is only psuedorandom.    Is there a library (specifically java) that generates random numbers according to random variables like environmental temperature, CPU temperature/voltage, or anything like that? Browse local copies of websites when possible (Firefox with slow internet connection) I am on a slow internet connection. I would like to use Firefox to load local copies of websites when it exists and is relatively recent (say one week).   An example usage would be browsing a lot of Wikipedia webpages, coming back to the same ones on a regular basis.  I would like to avoid having to manually specify to save single pages on each website I visit (which is what I guess I would need to do with a plugin like ). And I would also like to avoid having to switch between "work offline" or "work online".   Any suggestions? I want to calculate the derivative of $|x|^\alpha$ with respect to $x$, where $1&lt;\alpha&lt;2$.  $\frac{d|x|^\alpha}{dx}=\alpha|x|^{\alpha-1}\mathrm{sign}(x)$  Is it correct?  And is it true that  $\lim_{x \to 0} \alpha|x|^\alpha\mathrm{sign}(x)=0$?  What with the following derivative $|x|^\alpha\rm{sign}(x)$. I would be grateful for any hints. According to preliminary estimates, the project should be completed by 2020 at a cost of about RUB 40 bn.    or      According to preliminary estimates, the project should be completed by 2020 at the cost of about RUB 40 bn.    Are both options okay, or is the first better?   We are introducing the sum to the reader, so logically it should be the first option. But there's a lot of google-hits for "at the cost of" too. I'm new to Linux but not to app development. I come from a RealBasic on a Mac background.   I've accepted that I need to learn a new language to develop for Linux (since RB won't run on 64 bit distros and is a cloed proprietary language) but I want to make sure I pick the right one.   I've been looking at Python and was all set to go until I read that PyGTK is becoming obsolete with the introduction of GTK 3. Is this true? The Ubuntu developer website stills seems to recommend it with Glade for the UI but I want to make sure that I don't learn something that's on the way out! Is it possible to run three versions of Google Chrome? How might I run 3+ versions of google chrome so I can log into to 3 different google accounts.  More specifically I am looking to have separate icons in the dock for the 3 separate running applications.  How might I run duplicate and or copy versions of the same application.  The following  by Mark works for a lot of applications but not for Google Chrome.  It appears the following might work .