This is the question: The Fibonacci numbers are recursively dened by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n &gt; 1$. Prove that every fourth Fibonacci number is a multiple of $3$.  I've been stuck on this question for a while. If someone could help me out or at least help me start it, that would be great
(a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture. (b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical induction to prove your conjecture  I understand that for part a that all multiples of 3 of n are even. So F(0),F(3),F(6)... I just don't understand how to prove it.  For part B it is the same thing except multiples of 4  Please help, thank you!
How to avoid noisy renders in Cycles? I know that this question is a bit old or maybe overdone, but I'm running the latest edition of Blender and when I use the Cycles engine I get really fuzzy results and I can't find a simple, non-compromising solution to the quality.  I don't know exactly what I'm doing wrong here, but I'm sure others share my frustration.  Leaving everything in default settings, I started a new project and switched to the Cycles render.  I duplicated the cube and set colors for two Diffuse BSDF shaders, then went ahead and made walls and a floor all sharing the same diffuse shader.  I made two windows to emit blue light.  The screenshot has gaps between the walls because of the setup:  I'm aware that it's considered very incomplete and many have come to terms with some amount of noise, but is this normal?  Furthermore, with default settings or not, I can't change the amount of noise produced.  Supposedly, using "progressively refine" in the rendering tab should make it refine itself indefinitely, however Cycles only counts to 10 refines and then stops, creating a render which is a pixel-perfect match of the ordinary rendering method when it's completed.  I understand that there are tricks that can be used to reduce noise, but at the cost of extra processing time and usually small quality conflicts, like blurring.  My question essentially boils down to is this Cycles' current limit? Adding button into a Listview in WinForms Is there a way to add a button control to a cell in inside a ListView in a WinForms app? Is anti-gravity (i.e. repulsive gravity) possible in theoretical physics? Is  (i.e. repulsive gravity) possible in string theory?  I have read some articles about scientists making assumptions about the existence of anti-gravity, but is it possible in string theory? I recently received this rather amusing message while reviewing a user's first post:  Congratulations! This was only a test, designed to make sure you were paying attention. This post has already been removed, but thanks for taking time to leave feedback for the author.   What is the purpose of this test? How does the audit work? What happens if I fail? Why was the system expecting me to negatively review a good post, or positively review a bad post? A review is obviously an audit. Should I report it? Do review audits count toward the Reviewer badge? Do robo-reviewers dream of electric sheep? What should I do if a user posts sensitive information as part of a question or answer? Why is defragmentation unnecessary? If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph? Deriving Chinese Remainder Theorem from gcd Bezout identity We want to find the solution $x$ to the congruence system  $$\begin{align} x &amp;\equiv r_1 \!\!\!\pmod{\!m_1}\\ x &amp;\equiv r_2 \!\!\!\pmod{\!m_2}\end{align},\ \ {\rm where}\ \  \gcd(m_1, m_2) = 1$$  These can be rewritten as  $$\begin{align} x &amp;= r_1 + m_1 j\\ x &amp;= r_2 - m_2k\end{align}$$  for unknown integers $j, k$. So we set those two equations as equal and rearrange them to  $$m_1j + m_2k = r_2 - r_1$$  Now assume we perform $\text{egcd}(m_1, m_2)$, which gives us the solution to  $$m_1u + m_2v = \gcd(m_1, m_2)$$  Let the integer $\,h\,$ make $\,h \gcd(m_1, m_2) = r_2 - r_1$.  Then $\qquad hm_1u + hm_2v = h\gcd(m_1, m_2) =  r_2 - r_1 = m_1j + m_2k$  At this point I am lost because I don't think we can just assume  $hm_1u = m_1j$ and extract $j = hu$.   How do I get the values of $j$ or $k$ so I can get the value of $x$? How do I right align controls in a StatusStrip? Why do most C programmers name variables like this:  int *myVariable;   rather than like this:  int* myVariable;   Both are valid. It seems to me that the asterisk is a part of the type, not a part of the variable name. Can anyone explain this logic? I'm a simple minded Bayesian who feels comfortable in the cosy world of Bayes.   However, due to malevolent forces outside my control, I now have to do introductory graduate courses about the exotic and weird world of frequentist statistics. Some of these concepts seem very weird to me, and my teachers are not versed in Bayes, so I thought I'd get some help on the internet from those who understand both.   How would you explain the different concepts in frequentist statistics to a Bayesian who finds frequentism weird and uncomfortable?   For example, some things I already understand:   The maximum likelihood estimator $\text{argmax}_\theta \;p(D|\theta)$ is equal to the maximum posterior estimator $\text{argmax}_\theta \;p(\theta |D)$, if $p(\theta)$ is flat.  (not entirely sure about this one). If a certain estimator $\hat \theta$ is a sufficient statistic for a parameter $\theta$, and $p(\theta)$ is flat, then $p(\hat \theta|\theta)=c_1\cdot p(D|\theta)=c_1\cdot c_2\cdot p(\theta|D)$, i.e. the sampling distribution is equal to the likelihood function, and therefore equal to the posterior of the parameter given a flat prior.    Those are examples of explaining frequentist concepts to someone who understands Bayesian ones.   How would you similarly explain the other central concepts of frequentist statistics in terms a Bayesian can understand?  Specifically, I'm interested in the following questions:   What is the role of Mean Square Error? How does it relate to Bayesian loss functions? How does the criterion of "unbiasedness" relate to Bayesian criteria? I know that a Bayesian will not demand that its estimators are unbiased, but at the same time, a Bayesian would probably agree that an unbiased frequentist estimator is generally more desirable than a biased frequentist one (even though he would consider both to be inferior to the Bayesian estimator). So how does a Bayesian understand unbiasedness? If we have flat priors, do frequentist confidence intervals somehow coincide with Bayesian ones?  What in the name of Laplace is going on with specification tests like the $F$ test? Is this some degenerate special case of a Bayesian update on the distribution over model space?    More generally:  Is there some resource that explains frequentism to Bayesians?  Most of the books run the other way around: they explain Bayesianism to people who are experienced in frequentist statistics.     ps. I have looked, and while there are a lot of questions already about the difference between Bayesian and Frequentism, none explicitly explain Frequentism from the perspective of a Bayesian.   is related, but is not specifically about explaining Frequentist concepts to a Bayesian (more about justifying frequentist thinking in general).  Also, my point is not to bash frequentism. I really do want to understand it better Expanding electromagnetic field Lagrangian in terms of gauge field Inheritance in Java - creating an object of the subclass invokes also the constructor of the superclass. Why exactly? I have a question about inheritance in Java.  I have two classes A and B , and class B, inherits from A:  public class A {      public A() {          System.out.println("Hi!");      } }   public class B extends A {      public B() {          System.out.println("Bye!");      }       public static void main(String[] args) {          B b = new B();      } }   When I run program B, the output is:  Hi! Bye!      Question : why the constructor of class A is invoked, when I create and object of class B ?    I know that B inherits everything from A - all instance or class variables, and all methods, and in this sense an object of B has all characteristics of A plus some other characteristics defined in B. However, I didn't know and didn't imagine that when I create an object of type B, the constructor of A is also invoked. So, writing this:  B b = new B();   creates Two objects - one of type B, and one of type A.  This is getting interesting,   can somebody explain why exactly this happens? Can anyone explain reverse bias pn junction using only electrons as charge carriers In forward bias condition p-type semiconductor is connected to positive side and ntype semiconductor to negetive side and the electrons flow from n-type to p-type to cathode of battery.  But imagine the same situation when the situation is reverse biased so now p-type is connected to negetive terminal of battery and n-type to positive terminal of battery.  Now there is excess of electrons in n-region and it is connected to positive terminal of the battery.So electron can flow from n-type region to positive terminal of the battery due to potential difference created by battery whereas in p-type electrons can flow from one covalent bond to another as they do in forward biasing.  Only difference is,here electron move in opposite direction (from one covalent bond to another) i.e. from n-region to depeletion region whereas in forward bias they move from depeletion to n-region.Which shouldn't be a problem   Now in this manner current must flow regardless of biasing so what's the correction? Optimization paramters read-only mysql 5.7 database? What's the best way to configure a MySQL database that is only read-only, and will never have any modifications made to it?  1) I have increased my innodb_buffer_pool_size to 70% of available RAM  2) Increase query cache size to 40mb  Running against a load generator, after some time the database starts bottlenecking to only 1000 statements per second.  The configuration is 2-cores, 8GB ram 50GB SSD. The database contains a single InnoDB 13 GB table with the same exact query running on two indexed columns.  We are using a vertx server to hit the database. The symptom we see is that the web service performance degrades after 15-20 minutes of heavy load.  Where can the bottleneck be? What parameters should we optimize? $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property] Is $$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$ true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.  My tries  For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$  and thus $$[d_1,d_2](d_1,d_2)\mid n^2.$$  Question: How can I get $[d_1,d_2]\mid n$ from this?  For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow. Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks. Pass a PHP string to a JavaScript variable (and escape newlines) How should I structure my URLs for both SEO and localization? How can I see the history of stories I've read on Apple News? Sometimes I'll be interrupted in the middle of reading a story on Apple News and later when I open the app, it didn't save the story I was previously reading. I want to be able to resume reading the article, how can I find it again?