Prove the following identities involving Fibonacci numbers: $${F_n}^2 + {F_{n+1}}^2 = F_{2n+1}.$$ I am not sure how to work with the 2n's.
Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.  Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$  I get to:  $$f_{n+1}^2 + f_{n+2}^2$$  Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?  Thanks!
Accidently deleted boot option on Dell laptop I have a Dell laptop in which I have dual booted Ubuntu with Windows 10. Usually the laptop loads the Grub menu first.  Recently I had some problems with Windows Bitlocker: every time I open Windows I am prompted to enter the Bitlocker password. In trying to solve this, I accidentaly clicked Ubuntu in the "Delete Boot" option in the boot sequence. Now the laptop boots directly to Windows without first loading the Grub menu.  Luckily, when I restarted the system and clicked "add boot option" I found the efi folder in which the Ubuntu folder is present. In Ubuntu there is fwgrupx64.efi, grubx64.efi, grub.cfg, shimx64.efi, mmx64.efi, and BOOTX64.CSV. On clicking grubx64.efi it shows "Boot name not found".  How can I recover the Grub boot menu? Unit testing a method that calls another method I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ The exercise is: Prove that $-(-v)=v$ for every $v \in V$  Proof Suppose  $v \in V$ and $V$ is a vector space. Then $-(-v) \in V$ as result of the scalar multiplication property and $-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $ The desired result $-(-v)=v$ holds.  The solution manual gives the proof:  Proof   I just wanted to make sure that the way I did the proof isn't missing anything?Thanks How do programmatically create fields for content types, and ADD them to the the content type form Say I have this field, "map_description". I know I would this function to define the field:   $field = array(   'field_name' =&gt; 'map_description',   'cardinality' =&gt; 1,   'type' =&gt; 'text', ); field_create_field($field);   And I have this code which I'm not sure what it does but I'm told that I'll need it:   $instance = array(     'field_name' =&gt; 'map_description',     'label' =&gt; 'The map description.',     'bundle' =&gt; 'my_content_type',     'entity_type' =&gt; 'node',     'widget' =&gt; array(     'type' =&gt; 'text_textfield',  );  field_create_instance($instance)   Those two code bits are both in my install hook and run when I install the module. But while the fields are in fact created, I have to manually assign them to the content type via "manage fields", is there a way to have the fields automatically assigned to the content type? Making holes in a mesh I tried to find an answer to this a few years ago, but it seemed it wasn't possible then (I suspect it isn't now either! - but what the hell!)   I have a three monitor system, that I mostly use for remote desktop. However occasionally it would be useful to have remote desktop on two monitors and use the other monitor for the local machine.   I know it is possible to have a single monitor full screen, or to stretch an RDP session across monitors, but this loses things like snapping etc.   P.S. this is a duplicate of  thread, but that was a couple of years ago, I'm hoping things have changed. Running Windows 7  I'm trying to boost the startup speed of my computer and I'm going through "msconfig" and "services.msc" to disable some applications from starting up automatically when the computer turns on.  There are some good sites that document the entries found in the "Processes" tab of the task manager.  You can find out if they are safe to kill, who the manufacturer is, and what the process in question is used  for.  Are there any sites that list startup processes &amp;| services that are safe to kill?  For example, there is an entry in msconfig that just says "Computer Browser" - pretty vague, but seems like something to not kill.  On the other hand, there is an entry in msconfig that says "Extensible Authentication Protocol" - pretty specific and seems like it shouldn't be killed, but how can I know for sure?  Does anyone know if there are any websites documenting the "killable" items listed in both msconfig &amp; services.msc? How to apply custom headers/footers on pages with new chapters? I'm writing my final report in LaTeX for my bachelor's degree. I'm aiming at having a document similar to the .doc template used normally.   I want to have my customs headers and footers applied on pages when I'm starting a new chapter.  I'm using documentclass report with fancyhdr and lastpage packages.  My headers are defined like this:  \lhead{\myTitle} \rhead{\today} \cfoot{\thepage\ of \pageref{LastPage}}   My chapters are created using standard \chapter{title} macro.  On pages with new chapters, the only headers/footers I see is the standard page number at the bottom.  What should I do?  EDIT: I've finally gone with Peter Breitfeld's solution. I'Ve gone a step further and created my own chapter command:  \newcommand{\myChapter}[1]{     \chapter{#1}     \thispagestyle{fancy} }   Thanks for the help! What does the phrase "begging the question" really mean? And does it even matter if I use it correctly? Almost everyone just uses it as a synonym for "posing the question" these days. Energia/MSP-EXP432 Issue: Analog In Pin removing voltage from Voltage Divider In the circuit below, I want to measure the voltage going to the ADC (output from the voltage divider consisting of a fixed resistor and photoresistor) using an MSP-EXP432; however, the MSP-EXP432 "Analog In" pin is lowering the output voltage (i.e. the voltage going into the ADC from the voltage divider drops when the "Analog In" pin is connected).  I set the analog pin as an input in the code shown below in Energia.  Is there any way to fix this?    const int bitsOfADC = 14; // 14-bit ADC Launchpad //int maxVal = 2^N -1;  // 2^14-1 int maxVal = 16383; float FSV = 5.0; // Supply to 0804ADC float LaunchPadResolution = FSV/maxVal; int D7 = A13; int D6 = A14; int ADCInput = A6; boolean D7_State = LOW; boolean D6_State = LOW; void setup() {   delay(1000);   Serial.begin(9600);   pinMode(D7, INPUT);   pinMode(D6, INPUT);   analogReadResolution(bitsOfADC);   pinMode(ADCInput, INPUT); } Suggested edit review should show custom rejection reasons Somewhat similar to , but rather more limited: when Rejecting an edit, not only the count of custom rejection reasons but the actual text of the reasons should be shown, IMO. Just now I  on SO that tidied some code in a trivially correct and substantially superior way because it looked perfectly legitimate and I couldn't understand why anyone would bother with a custom rejection reason. As it turned out, it was code in a question, of course.  In this case, the rejection reason correctly identified the problem, and had I seen it, I would have been reminded of that aspect I had neglected to check. Generate uniformly distributed weights that sum to unity? It is common to use weights in applications like mixture modeling and to linearly combine basis functions. Weights $w_i$ must often obey $w_i ≥$ 0 and $\sum_{i} w_i=1$. I'd like to randomly choose a weight vector $\mathbf{w} = (w_1, w_2, …)$ from a uniform distribution of such vectors.  It may be tempting to use $w_i = \frac{\omega_i}{\sum_{j} \omega_j}$ where $\omega_i \sim$ U(0, 1), however as discussed in the comments below, the distribution of $\mathbf{w}$ is not uniform.  However, given the constraint $\sum_{i} w_i=1$, it seems that the underlying dimensionality of the problem is $n-1$, and that it should be possible to choose a $\mathbf{w}$ by choosing $n-1$ parameters according to some distribution and then computing the corresponding $\mathbf{w}$ from those parameters (because once $n-1$ of the weights are specified, the remaining weight is fully determined).  The problem appears to be similar to the  (but, rather than picking 3-vectors whose $ℓ_2$ norm is unity, I want to pick $n$-vectors whose $ℓ_1$ norm is unity).  Thanks! proof that $f'$ is continuous at $a$ Let $f:I \rightarrow \mathbb{R}$ derivable in set I. if $\lim x_n = \lim y_n =a \in I$ with $x_n \neq y_n$ and $\lim \frac{f(y_n)-f(x_n)}{y_n - x_n} = f'(a)$ proof that $f'$ is continuous at $a$  my attempt $x_n \neq y_n$ so by MVT $\exists z_n \in I$ like $x_n \geq z_n \geq y_n$ with $\frac{f(y_n)-f(x_n)}{y_n - x_n}=f'(z_n)$ for each $x_n$ and  $y_n$ then $f'(a) = \lim\frac{f(y_n)-f(x_n)}{y_n - x_n}= \lim f'(z_n)$   Using $\lim x_n = \lim y_n = a$ and $x_n \geq z_n \geq y_n \Rightarrow z_n \rightarrow a$  if $ \lim_{x\rightarrow a} f'(x) \neq f'(a)$ then $\exists \epsilon &gt; 0$ such $\forall n\in \mathbb{N}$, $z_n \in [x_n,y_n]$ $0 &lt; |z_n - a| &lt; \frac{1}{n}$ and $|f'(z_n) - f'(a)| \geq \epsilon$. So $z_n \rightarrow a$ but will do not have $\lim f'(z_n) = f'(a)$  so $\lim f'(z_n) = f'(a) = \lim_{x\rightarrow a} f'(x)$ when $z_n \rightarrow a$  and thats implies $f'$ continuous at $a$  my question is, that's correct ? if not, can you help me? thanks Which graphics formats can be included in documents processed by latex or pdflatex? My university's thesis guidelines require that each bibliographic entry appear in whole on a single page, rather than being split across two. How can I prevent (La)TeX breaking a paragraph across a page boundary? What can interfere with glossaries to prevent printing? I have the strange experience that from the combination      \glossarystyle{altlisthypergroup}     \clearpage     Printing acronyms     \printglossary[type=acronym]   only the debug message is printed, the glossaries not.  Even, the \printglossary leads to hanging. At the same time, in the text commands like \gls{} work as expected.  Any advice, what other package can interfere with glossaries? (It is part of a bigger project, the MWE works as expected.)    I changed to command line version from TeXstudio. I receive the output  makeglossaries DSc2 makeglossaries version 2.12 (2014-01-20) added glossary type 'main' (glg,gls,glo) added glossary type 'acronym' (alg,acr,acn) makeindex  -s "DSc2.ist" -t "DSc2.alg" -o "DSc2.acr" "DSc2.acn" This is makeindex, version 2.15 [TeX Live 2013] (kpathsea + Thai support). Scanning style file ./DSc2.ist.............................done (29 attributes redefined, 0 ignored). Scanning input file DSc2.acn....done (9 entries accepted, 0 rejected). Sorting entries....done (36 comparisons). Generating output file DSc2.acr....done (15 lines written, 0 warnings). Output written in DSc2.acr. Transcript written in DSc2.alg. Warning: File 'DSc2.glo' is empty. Have you used any entries defined in glossary 'main'? Remember to use package option 'nomain' if you don't want to use the main glossary.   What I do not understand, that if there are entries accepted, and sorted, why do I receive the warning about empty .glo file? And also, why the pdflatex hangs if I try to continue with using pdflatex again? Why does paper become translucent when smeared with oil but not (so much) with water? When I smear oil onto a scrap of paper and rub it in, the paper becomes quite translucent; but when I attempt the same with water it doesn't as much. Why? I am trying to identify a sci-fi movie from the 80's or 90's, with the following parameters:  Setting:    Scenes of a red desert planet   Culture:    Either people roam fairly solo and try to acquire resources to survive, or you live in a weird society where if a stranger is captured he needs to play in a game of fight to the death.  Mobility:      One foil character has a giant battle-bot-looking tank as his form of transportation through the desert. Protagonist has a car that has a plastic half bubble on top to look out from.   Robot:      The protagonist runs around in the beginning of the film with a female robot/android whose face is melted off after it has acid touch its face.(This scene happens next to the protagonist's rover)  Love interest:      Somehow protagonist comes across female protagonist in desert. She smells really bad and is hungry. They sleep outside, and he tells her to sleep away from him. He uses a thermal blanket(the first I ever saw in my life). She wakes up next to him, smelling horrible. He bathes her in some water hole they find(as if water were not scarce and they could just waste water.)   Pupa monsters:      Scene where they are in a building and they are trying to get out. They come across a room that is white, in which multiple giant, white, snot, pupa are hanging securely from the ceiling. Through curiosity, they disturb them, and now they have to get out of that room fast.  Boss Antagonist:      Pervert bionic man, that is actually attached to a giant robotic arm, as only form of movement. He is in black and has a pale bald head as his most human looking part. Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ is an open set, and if $E$ is finite dimensional then $\mathcal I$ is dense in $\mathcal B(E)$. It's not true that $\mathcal I$ is dense if we can find $T\in\mathcal B(E)$ injective, non surjective with $T(E)$ closed in $E$, since such an operator cannot be approximated in the norm on $\mathcal B(E)$ by elements of $\mathcal I$ (in particular $E$ has to be infinite dimensional).  So the question is (maybe a little vague): is there a nice characterization of $\overline{\mathcal I}^{\mathcal B(E)}$ when $E$ is infinite dimensional? Is the case of Hilbert space simpler?