Prove the following combinatoric identity:  $$\sum_{k=0, k even}^n {n \choose k}*2^k = \frac{(3^n)+((-1)^n)}{2}$$
I need to prove the following identity:  $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$  I know that -   $\sum_{k=0}^{n}{n \choose k}2^{k}=3^{n}$  but don't know how to continue.
With a macro such as:  \def\naive{na\"{\i}ve}   I find that the space which I would hope to follow it, is absent. So, with  the \naive approach   I get     the naïveapproach   in the output. How do I bring back the space? When a user without editing privileges suggests an edit to the post, a banner gets displayed until it is reviewed. Can the banner also provide a link to the Suggested edit? For example, I made an edit to a post on Unix&amp;Linux.SE. Can the banner also link to my ?     I know I can access the link through my Profile → Activity → Suggestions. But linking it directly on the face of the post would have immense benefits:  I have been a prolific editor on Ask Ubuntu (my first encounter with Stack Exchange Network) with a lot of Rep derived through Suggestions. I didn't get to know about these suggested edit links until very late (I guess I already had got about 400 edit suggestions approved by the time this came to my knowledge).  This means I didn't know about  when it mattered the most.  I never got to know why my suggestions got rejected because of my ignorance.  This would help reduce the barrier between Reviewers and Editors.  This would also help in case of Anonymous edits. Right now, there is no way an anonymous editor can access it unless the edit gets approved and he comes back to go through the Revision page of that particular post in order to access the Suggested Edit Review link. If the edit gets rejected - it's the end of the road for that anonymous editor. I just made an anonymous edit on one of the SE sites where I don't have an account and already feel like it's the end of the road for me since I just can't access the review link unless it gets approved.    Possibly also expand the banner to mention that they can also access the link to the suggestion via Profile → Activity → Suggestions. How do I set up my LaTeX document to enable me to cite from my BibTeX bibliography? This may be an overly simplistic question but please bear with me...  I have a BibTeX bibliography containing all of my references, set up using BibDesk. I am now trying to cite these references in my LaTeX documents. Following the advice on the LaTeX wikibooks page, I have included the following lines into my LaTeX document...  \bibliographystyle{plain} \bibliography{/Users/Graeme/Documents/References/BIBliography.bib}   ... where BIBliography is my .bib file, created in BibDesk.  Unfortunately, when I cite something in the document, using \cite{citekey} and typeset the document, the log tells me that there were undefined references. In the .pdf that is produced, there is a [?] where the references should be.  Am I doing something wrong? Do I need to store my .bib for in a particular location? I know there can be some errors that mean you need to typeset the document several times, but I have tried countless times to no avail.  I would be very grateful if someone could help out. Run 32bit applications on Windows 10 64 Bit I've got a Windows 10 Computer (home edition), which has 4GB of Ram and an i5 processor. I need to run an old 32 bit accounting programme. The vendor said it won't release a 64 bit version (although a 64bit version exists, it's missing key features from the older version).  The programme installs fine, but when I click on the programme icon, it says the programme won't work on a 64 bit system.   What options do I have?    I've done a fair bit of research, but it seems to run up dead ends.   1) WoW64     Apparently, this is supposed to let you run 32 bits on 64 bit Windows. However, I am not sure how to install it. I think it's for web servers only. However,  says it can work on Windows 7, but doesn't explain how to set it up (or if it will work on Windows 10).  2) XP mode     A 32bit Windows XP emulator. However, it seems to be for Windows 7 only. Also according to this , you need Windows Professional.   3) Virtual Box Machine    Let's you emulate Windows. Seems overkill for my purposes. Plus I would need to buy a 32 bit Windows license. I think it would run slowly with 4GB of Ram.  4) Partition drive into 32bit and 64 bit versions Apparently, you can use the free tool  to split the drive into two and then install separate 32bit and 64 bit versions.   Again, seems overkill and it looks like I would need buy a 32 bit Windows 10 license.   Are these my only options?     Images  Installation    First run (From the error message I guess it's a 16-bit program)   Clicking on programme Live refresh shows "modified" on tagged questions even when unmodified I want to know is there a way to render Shadows only in Blender's new EEVEE Render Engine. I want my objects in a separate collection and a ground plane on a separate collection(to receive shadows from the objects collection).I want to render the shadows only and not the objects. Is $\sqrt{64}$ considered $8$? or is it $8,-8$? Do Indian citizens need transit visa for Thailand if going through DMK? I am traveling from Chennai to Don Mueang International Airport, Bangkok, Thailand, and from there to Mandalay, Myanmar. In Bangkok, the layover time is 8 hours. Do I require transit visa? How do I render a Twig template and load a CSS file from a controller? Is this considered a good practice for rendering a Twig template and loading a CSS file from a controller?   DefaultController.php  namespace Drupal\acme\Controller;  use Drupal\Core\Controller\ControllerBase; use Drupal\Core\DependencyInjection\ContainerInjectionInterface; use Symfony\Component\DependencyInjection\ContainerInterface; use Drupal\Core\Template\TwigEnvironment;  class DefaultController extends ControllerBase implements ContainerInjectionInterface  {   /**   * @var Drupal\Core\Template\TwigEnvironment   */   protected $twig;    public function __construct(TwigEnvironment $twig)   {     $this-&gt;twig = $twig;   }    public static function create(ContainerInterface $container)   {     return new static(       $container-&gt;get('twig')     );   }    /**    * hello    * @param  string $name    * @return string    */   public function hello($name)    {      $template = $this-&gt;twig-&gt;loadTemplate(       drupal_get_path('module', 'acme') . '/templates/hello.html.twig'     );      $markup = [       '#markup' =&gt; $template-&gt;render([ 'name' =&gt; $name ]),       '#attached' =&gt; [ 'css' =&gt; [           drupal_get_path('module', 'acme') . '/assets/css/acme.css'         ]       ]     ];      return drupal_render($markup);   } }   acme.routing.yml  acme_hello:   path: '/acme/hello/{name}'   defaults:     _content: '\Drupal\acme\Controller\DefaultController::hello'     _title: 'acme Title'   requirements:     _permission: 'access content'   hello.html.twig  &lt;div class="acme-hello-text"&gt;   &lt;h1&gt;Hello {{ name }}!&lt;/h1&gt; &lt;/div&gt; MySQL cartesian product between two SELECT statements How to contact Canonical? So I am doing a College research project for my Computer Information Systems department and it is about alternative Linux server distribution for small businesses. From my findings I would really like for this research to actually take off into a Distribution project (Using Ubuntu as a base OS).   However I would like to rename the distribution as "Bizbuntu" or something related to that ("Biz" being shorthand for business) however I don't know if I can name the distribution without consent from Canonical.   So my question is do I have to contact Canonical about this, if so, does anyone know a mailing list or email that I can contact them about this? Using pre-compiled headers with CMake I picture light ray as a composition of photons with an energy equal to the frequency of the light ray according to $E=hf$. Is this the good way to picture this? Although I can solve elementary problems with the formulas, I've never really been comfortable with the idea of an object having or being related to a "frequency". Do I need to learn quantum field theory to really understand this? I have a LaTeX editing environment that does some assistive stuff such as auto-indenting, closing parentheses, etc.  However, one of the things that it does is unknown to me, and is difficult to search for. If I type an empty set of braces, it replaces them with  {}&lt;++&gt;   If I type two empty sets, for example when I'm setting up a fraction \frac{}{} what I actually get is  \frac\left\{ } \right\}&lt;++&gt;}&lt;++&gt;   Is this a problem? What does &lt;++&gt; do?  Edit: This is using Quickly read the last line of a text file? Assume that given three predicates are presented below: $H(x)$: $x$ is a horse $A(x)$: $x$ is an animal $T(x,y)$: $x$ is a tail of $y$ Then, translate the following inference into an inference using predicate logic expressions and prove whether inference is valid or not (for instance, using natural deduction): Horses are animals.  Horses' tails are tails of animals. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. My initial idea was to consider similar sentence such as &quot;w is a tail of a horse&quot; to form required inference, but it was not successful. Would be welcomed to hear your ideas about this task. What forces affect the value of my ETF? What apps to sync movie subtitles easily are available for Ubuntu? I have a subtitle file that is about 3 to 4 seconds behind. What tools can I use to automatically move all the times in the subtitle file so they are correct with the time of the movie? Something that I can for example just add the time I want to add and it will add this time to the synced times in the subtitle files without having to edit each line individually. Is Iron Man stronger than the Hulk? In Avengers: Endgame, when Hulk initially uses the new Infinity Gauntlet, it takes a tremendous effort from him (in addition to almost burning his entire arm), in order to snap his fingers and restore the world as we knew it. Yet at the end of the movie, when Iron Man uses the Gauntlet, it seems very easy for him to snap his fingers.  How was Tony able to wield the stones with such ease when Hulk was clearly struggling with it? Where did the disco repos go, or what is going on?