Prove: $$F^2_{n+1} - F_nF_{n+2} = (-1)^n$$ I do understand where does $$F^2_{k+2} - F_{k+1}F_{k+3} = $$ come from. But then it distributes as follows: $$=F_{k+2}(F_{k+1} + F_k)-F_{k+1}(F_{k+2} + F_{k+1})=$$ $$=F_{k+2}F_k - F^2_{k+1}=-(F^2_{k+1}-F_kF_{k+2})=-(-1)^k=(-1)^{k+1}$$ I see no pattern. Can someone explain this transition
I was looking at the decomposition of Fibonacci numbers using the definition of $F_n = F_{n-1} + F_{n-2}$, and noted the pattern in the coefficients of the terms were Fibonacci numbers. It appears to hold, and I believe it's true, but I haven't seen a proof for it. Does one exist?  $F_n = 1F_{n-1} + 1F_{n-2}$ $F_n = 2F_{n-2} + 1F_{n-3}$ $F_n = 3F_{n-3} + 2F_{n-4}$ $F_n = 5F_{n-4} + 3F_{n-5}$ etc  This can be generalized to $F_n = F_xF_{n-(x-1)}+F_{x-1}F_{n-x}$
How can the Next Generation's unfamiliarity with the Borg be explained after Archer's encounter? In  Starfleet learns about a culture that we (as audience) know as The Borg. According to TNG, the Federation doesn't seem to know anything about these beings when . Although Archer's crew doesn't learn the name of the Borg, it would be reasonable that the Federation would inherit the information about such a menacing enemy and it would also be reasonable to assume the crew of the Enterprise-D to be able to match their encounter with their databases' information.  Obviously, out-of-universe the explanation is that the scripts weren't written in chronological order, but how can the apparent obliviousness about the Borg be explained in-universe? Change the font of figure captions I would like to italicize my figure captions and make them smaller.  How can I do that? How to transfer between International T2 to Domestic 1B terminals in Mumbai? I will be arriving in Mumbai Terminal T2 on a Turkish Airlines international flight. I have a separately-purchased IndiGo ticket 5 hours later from Terminal 1B (Domestic). In the past I heard of a free bus to transfer.    Is that still the case and if not how do I get to Terminal 1B from Terminal 2? Upgrade Debian Jessie (8) to Buster (10) possible or need to go sequential in versions upgrades? Just started working with a Debian Jessie (8) in a production environnent. While presently deployed it is not seeing any heavy use yet. I was looking at the OS , and I was thinking about how convenient it would be to just upgrade it before is start seeing some more steady use in our environment and possibly making things more complicated with a later upgrade.  I was looking for a procedure to upgrade from Debian v.8 to v.10 but most documented  are sequential in order (e.g. 8 -> 9, or 9 -> 10).  My question: should I upgrade sequentially, or there is some proper documented way to skip a main version while upgrading? Is it possible to browse the internet from a laptop so that the web sites you visit think you are browsing from a mobile device?  Say I need to connect via a very low bandwidth connection and just using the mobile version of the site is sufficient?  Currently I'm using Vista and Firefox 3.5 How to pass javascript value to controller..? I want to pass the java-script value to controller.I am using repeat on VF page.  VF page      &lt;apex:repeat value="{!listAdj}" var="Adj" id="hrs"&gt;       &lt;td&gt;&lt;div class="requiredInput"&gt;&lt;div class="requiredBlock"&gt;&lt;/div&gt;     &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);" &gt;       &lt;/apex:inputField&gt;     &lt;/div&gt;&lt;/td&gt;             &lt;apex:inputField id="hours" value="{!Adj.Hours__c}" /&gt; &lt;apex:inputField id="decimal" value="{!Adj.Hours_Decimal__c}" onchange="setHidden5(this);"/&gt;              &lt;/apex:repeat&gt;   Javascript code  function setHidden5(element){     //split the id of the inputField by ':' and then use the 5th index in the array to exact position of the inputField in the apex:repeat.   var idArray = element.id.split(':');     var hiddenRep1 =document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":decimal");          var hiddenRep2 = document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours");       if(element.value.length &gt;=1){        document.getElementById("thePage:form:hrs:hrs:"+idArray[4]+":hours").disabled = true; **hiddenRep2.value=hidden1.value;**                }                             }     }   I want to pass the hiddenRep2.value to controller.How to pass that. Hidden2.value is same as hidden1.value + some addition subtraction.i dont know how to pass that value. Can someone me sort out this.  Thanks Here is a DOM-based XSS flaw reported by SalesForce Security Review team in our code:   RootComponent  &lt;aura:component extensible="true" abstract="true" access="global"                 description="Root template component"&gt;     &lt;!-- ... lot of other code ... --&gt;     &lt;aura:attribute name="label"                     type="String"                     access="global"                     description="Label of the form input element" /&gt;     &lt;!-- ... lot of other code ... --&gt;  &lt;/aura:component&gt;   SubComponent  &lt;aura:component access="global" extends="c:RootComponent"                  description="Child extension of RootComponent"&gt;     &lt;!-- ... lot of other code ... --&gt;     &lt;label&gt;{!v.label}&lt;/label&gt;     &lt;!-- ... lot of other code --&gt;  &lt;/aura:component&gt;   According to SFDC, "any app can extend SubComponent component and pass in  as a label and trigger a DOM-based XSS attack." The solution would be to "sanitize" label global attribute to avoid such attack. What is meant by sanitization of global attribute?  I tried to reproduce the so-called security flaw , but as label attribute is of type "String" , when we send it HTML, it is displayed as a string, not as interpreted HTML ... so I don't see where the security flaw could be :/  &lt;aura:application &gt;     &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;      &lt;c:SubComponent aura:id="TestXSS" label="DOM-based XSS attacks" /&gt;  &lt;/aura:application &gt;   JS Controller  ({     doInit: function(component, event, helper) {         var XssAttack = '&lt;img src=x onerror=alert(1) /&gt;';        component.find('TestXSS').set('v.label',XssAttack);     } })   Output    We have global attributes everywhere. Our components are packaged and meant to be extended by client implementation. Do we actually need to sanitize them? $1,000 is pronounced as "one thousand dollars". Reading from left to right, it seems like it would make more sense to write the value as: 1,000$. This way the pronunciation of the value follows the way it's written. Every other numerical value I can think of places the unit after the value:   1 kg 1 L 1 m, etc...   I'm not sure if this question fits in with the rest of the questions on this site. But I'd like to know why currencies are so different from other quantities. Not to mention the need to separate groups of three digits with commas. Are unequal groups a problem for one-way ANOVA? I have data for three unequal groups: $N = 44$, $N = 354$ and $N = 347$. Is it possible to compare all three groups running a one-way ANOVA or is the first group too small? How do you match only valid roman numerals with a regular expression? Thinking about , i decided I can't even create a regular expression that will match roman numerals (let alone a context-free grammar that will generate them)  The problem is matching only valid roman numerals.  Eg, 990 is NOT "XM", it's "CMXC"  My problem in making the regex for this is that in order to allow or not allow certain characters, I need to look back. Let's take thousands and hundreds, for example.  I can allow M{0,2}C?M (to allow for 900, 1000, 1900, 2000, 2900 and 3000). However, If the match is on CM, I can't allow following characters to be C or D (because I'm already at 900).  How can I express this in a regex? If it's simply not expressible in a regex, is it expressible in a context-free grammar? How do file permissions work? Slow BIOS bootup This is a new HP DV-7 laptop. When you turn it on SOMETIMES it will wait at 15 minutes on the press escape for menu and sometimes it will boot correctly.  IS this a hardware failure issue or is this a new for a BIOS update? Which year does Spider-Man: Homecoming take place in? Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$ I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$. Should spam posts be edited? I noticed that the posts of the recent spam surge on Ask Ubuntu were all edited by community members into something like     EDITED - REMOVED SPAM ANSWER   or similar.  While I understand why the users have done this and I too don't want to leave the spam content visible for longer than absolutely necessary, it made flagging more time consuming for me.  I had to check the revision for every of those possible spam posts to check if it is actually spam, instead of being able to flag directly from the review site. Spam flags carry a serious penalty, I'm not willing to add my spam flag without checking that it's really spam.  I see the benefit of editing spam, but it also causes some more effort for the later flaggers. Should we as a general policy edit out the content of spam posts, or should we deal with them solely by flagging? Computers on board Starfleet vessels are quite primitive compared to on board computers such as . At their best, they can decipher requests such as "Computer, delete that last sentence" and utter things such as "That crewman is no longer on board this vessel". Even real life computers will be capable of this within a decade or so (by some estimates). Even computers such as HAL 9000 are likely to be available within a few decades.  So, in a genre where intelligent computers are abound, why this strange absence? Did the Federation run into some Battlestar-Galactica-like problem with AI in the past and decide to keep on board computers decentralized?  But if that is the case, how can one explain Data and highly intelligent holographic characters? The technology and programming constructs clearly exist, yet Starfleet vessels continue to be manned by hundreds of people standing in front of consoles. $n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{0}^{n-1}\frac{365-b}{365}$. How many people would have to attend a meeting so that there is at least a $50$% chance that two people share a birthday?   So I set $\Pi_{0}^{n-1}\frac{365-b}{365}=.5$ and from there I manipulated some algebra to get $\frac{364!}{(364-n)!365^{n}}=.5\iff (364-n)!365^{n}=364!/.5=.....$   There has to be an easier way of simplifying this. My fish's native habitat is rice fields. My fish's native habitat are rice fields.   Which one is correct? I'm pretty sure it's the first, since 'is' modifies 'habitat,' but it still sounds weird... How do I check to see if a text file is empty or does not exist in c++? How do I pass a function as a parameter to in elisp? I'm trying to pass one method to another in elisp, and then have that method execute it.  Here is an example:  (defun t1 ()   "t1")  (defun t2 ()   "t1")  (defun call-t (t)   ; how do I execute "t"?   (t))  ; How do I pass in method reference? (call-t 't1)