\begin{equation} F_n=\frac{(\frac{1+\sqrt{5}}{2})^{n}-(\frac{1-\sqrt{5}}{2})^{n}}{\sqrt{5}} \end{equation}   Proof: Let n=1 thus, \begin{align*} F_1&amp;=\frac{(\frac{1+\sqrt{5}}{2})^{1}-(\frac{1-\sqrt{5}}{2})^{1}}{\sqrt{5}}\\ &amp;=\frac{\frac{1+\sqrt{5}}{2}-\frac{1-\sqrt{5}}{2}}{\sqrt{5}}\\ &amp;=\frac{\frac{1+\sqrt{5}-1+\sqrt{5}}{2}}{\sqrt{5}}\\ &amp;=\frac{\frac{\sqrt{5}+\sqrt{5}}{2}}{\sqrt{5}}\\ &amp;=\frac{\frac{2\sqrt{5}}{2}}{\sqrt{5}}\\ &amp;=\frac{{\sqrt{5}}}{\sqrt{5}}\\ &amp;=1 \end{align*} Suppose \begin{equation} F_k=\frac{(\frac{1+\sqrt{5}}{2})^{k}-(\frac{1-\sqrt{5}}{2})^{k}}{\sqrt{5}} \end{equation} Also $F(k+1)=F(k)+F(k-1)$ \begin{align*} F(k)+F(k-1)&amp;=\frac{(\frac{1+\sqrt{5}}{2})^{k}-(\frac{1-\sqrt{5}}{2})^{k}}{\sqrt{5}}+\frac{(\frac{1+\sqrt{5}}{2})^{k-1}-(\frac{1-\sqrt{5}}{2})^{k-1}}{\sqrt{5}}\\ &amp;=\frac{(\frac{1+\sqrt{5}}{2})^{k}+(\frac{1+\sqrt{5}}{2})^{k-1}-(\frac{1-\sqrt{5}}{2})^{k}-(\frac{1-\sqrt{5}}{2})^{k-1}}{\sqrt{5}}\\ &amp;=\frac{((\frac{1+\sqrt{5}}{2})+1)(\frac{1+\sqrt{5}}{2})^{k-1}-((\frac{1-\sqrt{5}}{2})+1)(\frac{1-\sqrt{5}}{2})^{k-1}}{\sqrt{5}}\\ &amp;=\frac{((\frac{3+\sqrt{5}}{2}))(\frac{1+\sqrt{5}}{2})^{k-1}-((\frac{3-\sqrt{5}}{2}))(\frac{1-\sqrt{5}}{2})^{k-1}}{\sqrt{5}}\\ &amp;=\frac{((\frac{1+\sqrt{5}}{2})^{2})(\frac{1+\sqrt{5}}{2})^{k-1}-((\frac{1-\sqrt{5}}{2})^{2})(\frac{1-\sqrt{5}}{2})^{k-1}}{\sqrt{5}}\\ &amp;=\frac{(\frac{1+\sqrt{5}}{2})^{k+1}-(\frac{1-\sqrt{5}}{2})^{k+1}}{\sqrt{5}}\hfill  \end{align*}  I was told there was a mistake but I feel this is right can I get clarification on my proof if it seems right?
May someone help me? I am trying to use induction to prove that the formula for finding the $n$-th term of the Fibonacci sequence is:   $$F_n=\frac{1}{\sqrt{5}}⋅\left(\frac{1+\sqrt{5}}{2}\right)^n-\frac{1}{\sqrt{5}}⋅\left(\frac{1-\sqrt{5}}{2}\right)^n.$$  I tried to put $n=1$ into the equation and prove that if $n=1$ works then $n=2$ works and it should work for any number, but it didn't work. I need to prove that this formula gives the $n$th Fibonacci number.
In PHP (>= 5.0), is passing by reference faster? In PHP, function parameters can be passed by reference by prepending an ampersand to the parameter in the function declaration, like so:  function foo(&amp;$bar) {     // ... }   Now, I am aware that this is not designed to improve performance, but to allow functions to change variables that are normally out of their scope.  Instead, PHP seems to use Copy On Write to avoid copying objects (and maybe also arrays) until they are changed. So, for functions that do not change their parameters, the effect should be the same as if you had passed them by reference.  However, I was wondering if the Copy On Write logic maybe is shortcircuited on pass-by-reference and whether that has any performance impact.  ETA: To be sure, I assume that it's not faster, and I am well aware that this is not what references are for. So I think my own guesses are quite good, I'm just looking for an answer from someone who really knows what's definitely happening under the hood. In five years of PHP development, I've always found it hard to get quality information on PHP internals short from reading the source. nameref (hyperref) evaluating counter instead of section name What's the proper way to reference a section with a counter in the title? The TOC generates correctly, but the cross references aren't so lucky.  MWE:  \documentclass[letterpaper, 12pt, twoside]{exam}  \usepackage{hyperref}  \newcounter{KSRLessonNumberCounter} \setcounter{KSRLessonNumberCounter}{1} \newcommand{\KSRSection}[2][\value{KSRLessonNumberCounter}]{         \setcounter{KSRLessonNumberCounter}{#1}         \section*{Lesson \arabic{KSRLessonNumberCounter}: #2}         \addcontentsline{toc}{section}{Lesson \arabic{KSRLessonNumberCounter}: {#2}}         \stepcounter{KSRLessonNumberCounter} } \newcommand{\KSRSubsection}[1]{         \subsection*{#1}         \addcontentsline{toc}{subsection}{#1} } \newcommand{\KSRLessonNumber}{ }  \begin{document}  \KSRSection{Sample}  This is the first lesson. The second lesson is \nameref{sec:blah}.  \KSRSection{Blah} \label{sec:blah}  This is the second section  \KSRSection{Lorem Ipsum}  Dollars are Set in \nameref{sec:blah}.  \end{document} I recently came across an IE7 only bug that I thought I'd share so when I come to this site 6 months from now to figure out the same thing, I'll have it on hand.   I believe the easiest way to recreate this bug would be the following html in a page with a declared doctype (it works correctly in "quirks mode" / no-doctype):  &lt;div style="overflow: auto; height: 150px;"&gt;     &lt;div style="position: relative;"&gt;[...]&lt;/div&gt; &lt;/div&gt;   In IE7, the outer div is a fixed size and the inner div is relatively positioned and contains more content (assuming the inner div causes an overflow). In all other browsers, this seems to work as expected.  Screenshot: Where can I find information about upcoming special issues in a field of study? How can I find information about upcoming special issues related to a special field of study? There are good websites for upcoming conferences but I could not find any place to search for special issues. What is the preferred way to find focused control in WinForms app? Can this identity be derived from the binomial theorem?  $$n(n+1)2^{n-2} = \sum_{i=1}^{n}  i^2 \binom{n}{i}$$    I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it by $4$ in order to get $2^{n-2}$. Reproducible LaTeX builds - compile to a file which always hashes to the same value Prove p-adic valuation is completely additive for prime p Force MacBook to drain the battery while connected to charger I leave my MacBook Pro on my desk nearly all the time, connected to the charger with the lid closed, connected to an external display.  I've heard this is bad for the battery health, and everyone in my office have to replace swollen batteries every couple of years. I want to regularly discharge the battery, per , but I don't want to physically unplug it because that temporarily turns off the external monitor, and then I have to reconnect the monitor.  So I am looking for a software method of disabling the power input, to force the laptop to discharge its battery to 20% and then re-enable the power input. What is the name of the genre of games similar to Jetpack Joyride? I want to cd into /var/named but it gives me a permission denied error, and when I want to use sudo to do this I am not permitted. What is the technical reason for this, and is it possible to do this some other way? I have a financial transaction that looks something like this (pseudo):  begin     select record for update     update record(same record as selected)     insert another record into another table commit   If I call SELECT ... FOR UPDATE within the transaction, will the lock be released for the specific rows only when the transaction is committed ? Going to any question on the site brings an error page Why is absolute value  function   not a polynomial?   I need  a clear  answer to this question please,?  Why couldn't we consider absolute value function as a polynomial? Cartoon: 4 kids transform into video game characters to save the world from a hacker and an evil program There are 4 friends: 2 girls and 2 boys who play video games. A virus program escapes and, with the help of a programmer boy, transforms the world into a video game. But a good program in a white robe turns the four of them into video game characters to save the world.  The evil virus mostly looked like The Tick from the cartoon "The Tick". Big strong guy with muscles but in red spandex. Little head with antenas, BIG chin and evil mustache.  One hero is the typical blonde girl who can scream loudly with a microphone or headphones. The other chick was shorter, more a grumpy type and with her big metal boots she was able to rock / shake the ground. There was also a funny white boy with red hair, he was a cyber archer. And their black-skinned friend, a typical tech guy with maybe one laser eye.  The transformed world has always had a theme: a swamp, a volcano,  space, a cyber city, a sewer, once a pinball machine, and so on. For some game points they were able to summon different vehicles, weapons and reinforcements/buffs.  The story is made up of the usual repetitive events: every day the hacker guy transforms the world or just the city with a wicked program. Summoning new monsters and obstacles for stopping the good guys to defeat him.  But our heroes always win and the evil program comments what a loser he is and the guy get frustrated.  (In the end, they convince the hacker guy that it just using him and he chose the good side then they wiped out the evil program forever.) - NOPE XD  Not Code Lyoko, ReBoot, Linkers, Cyberchasers. There is no 3D; it was fully hand drawn. I think it was American, but was definitely not Japanese and there was no anime-like stuff. I was young when I saw it, i'm live in a foreign country Hungary. So i dont know when was it released, but maybe before 2005. The main group maybe has something "force" in their name. Cartographic techniques for symbolizing routing data Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. Equation number in aligned environment after last line I'm using aligned inside align as follows   \documentclass{article} \usepackage{amsmath} \begin{document} \begin{align} x =&22+2 \nonumber\\   =& \begin{aligned}[t]     &5+3\\     &-3+4\\     &+15-0   \end{aligned} \end{align} \end{document}   I want to have just one equation number after the very last line.  However, I'm getting the number at the beginning of the first split line (after the second = ),    and I cannot figure out how to move this equation number to the bottom. Any help will be appreciated. How to get text under mouse in any open window Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$ Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$ Subdivision Surface not making things smooth as they are in the tutorial I am following