write the most simple algebraic proof you can, that means no integrals and no exponential generating functions if possible.  I have tried to solve it for close to 2 hours. what I got to is almost nothing but here it is : $1/2 * \mathrm{2}^{2n+1}=\mathrm{2}^{2n}=\mathrm{4}^n=\mathrm{(2+2)}^n= \sum_{k=0}^n\binom{n}{k}\mathrm{2}^k\mathrm{2}^{n-k}=?\sum_{k=0}^n {n+k \choose n}\mathrm{2}^{n-k}  $  and that's it. this is the first time I tried to use symbols so if something goes wrong, I am terribly sorry.
I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me
I've already read all the pages in this site trying to find the answer to my problem but no one seems to be the right one form me...  First I explain you the kind of data I'm working with...  Let's say that I have an array vector with several names of city, one for each of 300 users. I also have another array vector with scores response to a survey of each user or a continuous value for each user.  I would like to know if exist a correlation coefficient that compute the correlation between these two variables so, between a nominal and a numeric/continuous or ordinal variables.  I've searched on the Internet and in some pages they suggest to use the contingency coefficient or Cramer's V or Lambda coefficient or Eta . For each of this measure the just say that they could be applied for such data in which we have a nominal variable and interval or numerical variable. The thing is that searching and searching, trying to understand every one of them, sometime is written or watching the examples that they are reasonable to use them if you have dichotomous nominal variable, except for Cramer's V, other time is not written any requirement for the type of data. A lot of other pages say that is right to apply regression instead, that is right, but I would just simply like to know if there is a coefficient like pearson/spearman for this kind of data.  I also think that is no so properly to use Spearman Correlation coeff since the cities are not sortable.  I have also built the function of Cramer'sV and Eta by myself (I'm working with Matlab) but for Eta they don't talk about any p-value to see if the coefficient is statistically significant...  In the matlabWorks site there is also a nice toolbox that says to compute eta^2 but the kind of input it needs is not understandable.  Is here someone that have done a test like mine? If you need more detail to understand the kind of data I'm using just ask me and I'll try to explain you better. Why does shell ignore quoting characters in arguments passed to it through variables? Evenly spaced menu items on Toolbar I'm having trouble distinguishing between the concepts of scedasticity and stationarity. As I understand them, heteroscedasticity is differing variabilities in sub-populations and non-stationarity is a changing mean/variance over time.   If this is a correct (albeit simplistic) understanding, is non-stationarity simply a specific case of heteroscedasticity across time? Consecutive spoiler blocks (>!) are shown as blockquotes The markdown &gt;! a  &gt;! b  &gt;! c  Will be shown as a blockquote with ! in front of it:   a    b    c  Instead of a spoiler for 3 paragraphs (or 3 spoiler blocks, whatever). Can only move object origin in object mode, cant move the mesh I can only move the origin point of the mesh that I selected in object mode, but not the whole mesh Did I set anything up by accident? elementary row operation change the eigenvalue of a matrix My question is simple, why do we believe  to be absurd? Is the Ultra-violet catastrophe incorrect or is it only because we can not create or know of a perfect emitter? I am a bit confused as to why Quanta was necessary. In  thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that  $$f(f(x)) = f(x) + x$$  (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is the  $\frac{1+\sqrt{5}}{2}$).    Having read about  shortly before though, I came up with this train of thought:  $$f(f(x)) = f(x) + x$$ $$\Leftrightarrow f^2 = f^1 + f^0$$ $$f^2 - f - f^0 = 0$$  where $f^n$ denotes the $n$'th iterate of $f$.  Now I solved the resulting quadratic equation much as I did with plain numbers  $$f = \frac{1}{2} \pm \sqrt{\frac{1}{4} + 1}$$ $$f = \frac{1 \pm \sqrt{1+4}}{2} = \frac{1 \pm \sqrt{5}}{2}\cdot f^0$$  And finally the solution $$f(x) = \frac{1 \pm \sqrt{5}}{2} x .$$  Now my question is: *Is it somehow allowed to work with functions in that way?** I know that in the above, there are denotational ambiguities as $1$ is actually treated as $f^0 = id$ ... But since the result is correct, there seems to be some correct thing in this approach.  So can I actually solve certain functional equations like this? And if true, how would the correct notation of the above be? How to simulate traceroute using ping? I am learning computer networks and looking for a way to simulate traceroute using the ping command. I think it has something to do with ttl, but I don't know how to develop that further. How to remove local (untracked) files from the current Git working tree Parsing Command Prompt Output I am retrieving disk information via JavaScript with the wmic statement, wmic logicaldisk get freespace,name,size,volumename, which produces the following output, but in a single string.    "FreeSpace      Name  Size           VolumeName  " "560232755200   C:    999526756352   System      " "999369699328   D:    999558213632   SSD         " "1511570386944  E:    8001545039872  Get         " "4620751712256  F:    8001545039872  BR          " "788449492992   G:    4000650883072  Seen        " "2296009408512  H:    4000768323584  Seen 2      " "3594248679424  I:    8001545039872  2160        " "3507750227968  J:    8001545039872  1080        " "945300619264   K:    999625322496   Trailers    "   I parse the above output with the following JavaScript.  window.onload = function(){   const cp = require('child_process')   let drives   cp.exec('wmic logicaldisk get freespace,name,size,volumename', (error, stdout)=&gt;{     drives = stdout.trim().split('\r\r\n')       .map(value =&gt; value.trim().split(/\s{2,0}/))       .slice(1)   }) }   That JavaScript already produces my desired output. It produces an array of nested arrays. Each row of information in the command prompt output corresponds to a nested array. Each nested array has four values that correspond to the originally queried data points.  [   ["560232439808",  "C:", "999526756352",  "System"  ]   ["999369699328",  "D:", "999558213632",  "SSD"     ]   ["1511570386944", "E:", "8001545039872", "Get"     ]   ["4620751712256", "F:", "8001545039872", "BR"      ]   ["788449492992",  "G:", "4000650883072", "Seen"    ]   ["2296009408512", "H:", "4000768323584", "Seen 2"  ]   ["3594248679424", "I:", "8001545039872", "2160"    ]   ["3507750227968", "J:", "8001545039872", "1080"    ]   ["945300619264",  "K:", "999625322496",  "Trailer" ] ]   I am wondering whether or not there is a way to similarly parse the information, but with command prompt instead? Is there any possibility to find my father's lost iPhone? My father has lost his iPhone 6 which was not signed in using an iCloud account. Is there any possibility it can be found? In attempting to wrap my mind around the basic vocabulary, concepts, and methods of philosophy, I find myself wondering what the difference is between a philosopher and a theologian.   can have two meanings: 1. Theology is a rational study of the existence of God/gods and the nature of religious ideas. 2. Theology is simply a study of a particular religion (or all religions), really more the practice than the theory, but maybe a mixture.  It seems from an immediate reading that definition 1 would lead to the conclusion that a theologian is a philosopher (perhaps that theology is a branch or subset of philosophy), but definition 2 appears to be rather unphilosophical in nature.    So, how does philosophy answer this question?   Is theology considered a proper subset/branch of philosophy overall? Does it depend entirely on which of the above 2 definitions of theology fits a particular theologian as to whether he is a philosopher or not?   I think this answer is simple and fairly self-evident, but somewhere in my gut I feel like I'm missing something.  Is there a treatment of this question in the literature?    Edit: the answers so far seem to want to redefine theology or other terms - that is fine, but please be clear that you are redfining them, and provide, if possible, sources for why the term should be different than given (support from historical works, other sites, etc.). How to import a jar in Eclipse I have a chain object. And i have a curve it follows. I got it to work for following curve, but chain object is not circles but both parts are more like stretched circles (ovals). Object i made with 2 toruses one rotated 90 degrees and then joined them together. I tried single torus too. But same issue. on a curve it just looks stretched oval. What can be the issue?   Thanks At the start of the movie it says:      "A long time ago in a galaxy far, far away...."   Does this mean that the entire Star Wars saga takes place in a single galaxy? To my understanding a galaxy isn't very big compared to the universe itself. Star Wars has many different sentient races, and the way they are presented suggests that most planets are host to only one of them (some exceptions like naboo). This strongly leads me to believe there should actually be more galaxies in the Star Wars universe (hehe).   Another thing, I think it's safe to assume that even if the entire saga takes place in one single galaxy, their researchers can observe other galaxies (I think we started doing that in the 1600s, and they are a lot more technologically advanced than we were back then). When Obi-wan is looking for Kamino (the rain planet with the cloners) the librarian says something like "If it's not in our records, it doesn't exist", so she must be convinced that the entire universe has been explored.    If the entire universe has been documented, why wouldn't they go to those other galaxies? The technology seems to be advanced enough to hop to another galaxy. Is there any canon material that suggest that traveling between galaxies is done? If not, is there a canon explanation why they don't? Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.    I have tried to using the definition but failed. Can someone help me please. Insert distinct data into Combobox List GOTO still considered harmful?