Solve $T(n) = 1 +\sum_{i=0}^{n-1}T(i)$ For the recurrence defined by  $$T(n) = 1 +\sum_{i=0}^{n-1}T(i)$$  Apparently $T(n) = 2^n$ .. but I cannot see it. This recurrence pops up during analysis of the . I keep looking to find a viewpoint to look at this problem where the factors of 2 pop out ..
How do you solve a recurrence with a summation function inside: $t(n) = 1 + \sum\limits_{j=0}^{n-1} t(j)$ Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$
Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I think $FK$ is pretty close to the free product of $F$ and $K$; is $FK \subsetneq F*K$? Thank you! Text Distortion Effect (like it is written on a page) I want to display a text like it is written on some surface like notebook page or wall via Photoshop CS6. I added two example.   This image seems like image distortion:    And this image looks like text distortion:    I tried this with distort and skew transformation but I'm unable to get this much clear &amp; real effect. What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography? What is the purpose of the Character Table in style files? What is the reason to insert the code below at the beginning of a style file? For example, the code below can be found on lipsum.sty.   %% \CharacterTable %%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %%   Digits        \0\1\2\3\4\5\6\7\8\9 %%   Exclamation   \!     Double quote  \"     Hash (number) \# %%   Dollar        \$     Percent       \%     Ampersand     \&amp; %%   Acute accent  \'     Left paren    \(     Right paren   \) %%   Asterisk      \*     Plus          \+     Comma         \, %%   Minus         \-     Point         \.     Solidus       \/ %%   Colon         \:     Semicolon     \;     Less than     \&lt; %%   Equals        \=     Greater than  \&gt;     Question mark \? %%   Commercial at \@     Left bracket  \[     Backslash     \\ %%   Right bracket \]     Circumflex    \^     Underscore    \_ %%   Grave accent  \`     Left brace    \{     Vertical bar  \| %%   Right brace   \}     Tilde         \~} %%   What is it for? I am running an HP pavilion dv6000 dual boot win7 and Ubuntu 12.04. (well, up until today).  After a reboot, the boot process drops to the BusyBox shell and I end up at the prompt:  BusyBox v1.18.5 (Ubuntu 1:1.18.5-1ubuntu4) built-in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs)   I've been researching others who have had this same problem, but haven't been able to find any of those solutions to work for me.  I tried the method described , and after the final command   mount -t ntfs-3g /dev/sda1 /root -o force   it does nothing and gives me another (initramfs) prompt.  I can boot to a live CD (USB) and get to a terminal, but it doesn't seem to do much good, as I can see the /dev/sda1 in the ls command, but it doesn't recognize it when I try to cd to it.  One more question: using the command fdisk -l how can I tell which mount point (sda1/sda2) is my windows partition and which one is Ubuntu? Where is composition tab in QGIS Composer to change page size? I am trying to change the page size in the QGIS 3.0 Composer but I am not able to see the Composition tab.  I have right clicked an empty space in the toolbar but did not see anything about composition.  I am using Windows 7.  I tried looking at the QGIS docs but they are still in testing. How does water evaporate if it doesn't boil? When the sun is out after a rain, I can see what appears to be steam rising off a wooden bridge nearby. I'm pretty sure this is water turning into a gas.  However, I thought water had to reach 100 degrees C to be able to turn into a gas.  Is there an edge case, for small amounts of water perhaps, that allows it to evaporate? How does Azkaban's management work? I was under the impression that only Dementors are guarding the prison of Azkaban and are only there to keep the inmates on their cells, though after reading again the following passage it put me into thoughts:     'One night, when (the Dementors) opened my door to bring food, I   slipped past them as a dog. It's so much harder for them to sense   animal emotions that they were confused ... I was thin enough to slip   through the bars. I swam as a dog back to the mainland. I journeyed   north and slipped into the Hogwarts grounds as a dog. I've been living   in the Forest ever since.'       Prisoner of Azkaban; chapter 'The Servant of Lord Voldemort'   So Sirius says that he escaped from his cell when the Dementors brought him food.... I say what??? Dementors dont obey anyone, not even the commands of the Ministry of Magic.  So the actual question is, How is the Azkaban prison managed? e.g. who cooks the food? the inmates have blankets, who is washing these blankets etc etc In the sentence "the best choice would be...", what does "would" mean in the following sentence?     Questioner: Is this sentence correct?   'Finished eating, we went to the zoo'.      Answer: I'm afraid that is not correct. The phrase 'Finished eating' is a participle phrase with a passive meaning and you need an active meaning here. The best choice would be 'Having finished eating...'. I am working with a basic  which is a popular modal used on many sites. I noticed that the default template comes with both a close button on the bottom, as well as an x on the top, these two buttons essentially having the same purpose. Are the two redundant and the close button completely unnecessary and just taking up extra space? Or, is having multiple usage options more user friendly?   To make this even more fun, suppose I throw in a Save button:   Is it now clearer that if you don't press save and just press the x, your changes will be canceled, or is there now more of a need for a cancel button for users who are scared that a mistake will become permanent? Update: To clarify for those who feel this question is similar to  - That question is asking if a clickable background is sufficient to close a popup. I'm using buttons and it is just a matter of how many buttons to use. (I do also have a clickable overlay and ESC key enabled, but that is irrelevant to my question) The set of real numbers and power set of the natural numbers I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$ Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$   I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$  We have $13\mid n^{13}-n$, by Fermat's Little Theorem.  We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is odd.  And the numbers $5$ and $7$, how to proceed? I know there is a difference between assert and verify in Junit; the assert results in a runtime exception, causing test method execution to abort. To overcome this problem, we can have verify which will make test case fail, but allowing test execution to proceed rather than getting aborted.  I doubt there is any such built in approach in testNG. I do understand that best practices say, there should not be multiple asserts in same test case.  But, in practical world, we do come across scenarios where we need to have multiple checkpoints in one test case and we don't want our test method to abort because of one check point failure (test execution should proceed despite failure).  How can we use a soft assert mechanism, similar to verify, which will not abort test method when there is a checkpoint failure.  Example:  @Test(dataProvider = "getMailUrl") public void testMail(email,subject); Assert.assertFalse(subject.isEmpty);//1st checkpoint Assert.assertNotNull(email); //2nd checkpoint Assert.assertEquals(email, "feedback@yoursite.com"); //3rd checkpoint   Now, I want test execution to proceed, even if the 1st checkpoint (validation of subject being not empty) fails. Definitely we will be not be using assert (hard assert). We will need concept of soft assert similar to verify (facilitated by Junit).   I understand, we have to build a customListener implementing TestNG interfaces or extending existing listeners such as TestListenerAdapter. I am still struggling - how can we do this? Often in my papers I want to mention theorems in the introduction that are introduced in more detail and proved in a later section.  I'd like to include the statement of these theorems twice over, each time numbered as the theorem would naturally appear in its later section.  My current solution (using amsthm) is, e.g.  %(in the introduction) \newtheorem*{thm:associativity}{Theorem \ref{thm:associativity}} \begin{thm:associativity} Lorem ipsum ... \end{thm:associativity}  %(in a later section) \begin{thm} \label{thm:associativity} Lorem ipsum ... \end{thm}      Can anyone think of a cleaner solution? What would be better practice when giving a function the original variable to work with:  unsigned long x = 4;  void func1(unsigned long&amp; val) {      val = 5;             } func1(x);   or:  void func2(unsigned long* val) {      *val = 5; } func2(&amp;x);   IOW: Is there any reason to pick one over another? Hovering over featured moves the questions down a bit When I hover over featured tab, both on the front page and in questions, the questions move down by one pixel. Here's a small screen capture demonstrating the issue:        I see it in Chrome 34 and Opera 20, but I don't see it on IE11 or Firefox 28. Storing Images in DB - Yea or Nay? Add folder name to beginning of filename How to format a USB or external drive? Using Ubuntu 11.10, how do I format an external or USB drive? I don't find an option to 'format' when I right click. In the new version, the drives don't come up on the desktop, but instead come on the Launcher.