So I am trying to prove  $$\sum_{i=0}^n{nCi×F_i} = F_{2n}$$ Such that $$nCi = \frac {n!}{i!×(n-i)!}$$ And $F_i$ is the ith value of the fibonacci sequence such that $F_0 = 1$ and $F_1 = 1$  I have stared at this problem maybe 3 hours so far and I have found so far $$\sum_{i=0}^{n/2}{(n-i)Ci} = F_{n+1}$$   and that   $$F_{2n} = F_{n+1} × F_n + F_n × F_{n-1}$$   but so far I haven't quite found a way to use these patters for a proof. Am I on the right track? If not could anyone spare a student a hint? :).  Thank you  Also I apologize for the poor format of my math expressions, I am still new to the site and I couldn't get any syntax I personallu knew to format.
I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing?
Using indefinite articles before adjectives: "Let's have a good breakfast" Let's have breakfast    Let's have a good breakfast      I've learned that indefinite articles aren't used before uncountable nouns, so why is a used before the adjective "good" here?    Is there an exception to the rules or is it wrong? When "Questions that may already have your answer" (similar questions) appear as a right column, it overflows and overlaps the footer.  (shown in the figure)        This happens on all SE network sites on which I have asked questions.  Also this happens to other columns also like "How to Format". We are given with  $$S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0  \}$$ Now observe that $-\sqrt n\leq x\leq\sqrt n $ and similarily, $-\sqrt n\leq y\leq\sqrt n $. This means that $-\left \lfloor{\sqrt n}\right \rfloor\leq x\leq \left \lfloor{\sqrt n}\right \rfloor$ and  $-\left \lfloor{\sqrt n}\right \rfloor\leq y\leq \left \lfloor{\sqrt n}\right \rfloor$. Through this result one can easily write code and figure out the cardinality of set $S_i$ but I was wondering whether an expression can be found that would allow one to directly compute $|S_i|$. Unbiased Estimator for a Uniform Variable Support Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $.  Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be unbiased.  I thought about simple estimator $ \hat{\theta} = \max \left( x_i \right) $.  Based on simulation it is not biased, yet I couldn't show it analytically.  Could anyone, please, show it is unbiased?  BTW, I could easily find another, easy to prove, unbiased estimator, $ \hat{\theta} = 2 \mathrm{mean} \left( {x}_{i} \right) $ reason for exec in wrapper scripts I have a huge amount of 2D-coordinates, associated with a value, e.g.:    x   |   y   | value 27.50   52.15   12.51 61.83   13.32   57.56 36.23   21.83   41.73 40.46   85.67   25.20 ...   The data is not tabular and I Want the points between two data-points to be interpolated in some way (which way is not really clear, yet)  I want to preset the data as heatmap like this:   Is there any ready-to-use package for PSTricks or TikZ to do it? I want to have a vector with a superscript:  \vec{o}^t   But this gives very bad results: the t is overlapping with the vector arrow. How do I fix this (and why does this give bad results in the first place) ? Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous? Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics.  I need to prove: $$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$ Can you please help me Tried to add a comment to a post today, which ended up looking rather weird. What I had typed was:  On windows I believe you need to use `\` instead of `/`   As you can see in the first comment to this question, I don't exactly end up with what I thought I would... What's going on here?  Tried various combinations of double \\ and double `` (which seems to be impossible to have in an inline code thing...?), but couldn't get a good result. Ended up putting a space after the slash, which at least made it readable. Problem then was that the space between the inline code and the word "instead" seems to disappear O.o How to prevent my screens from turning off, when the screen is locked? I would like to have my screen fully powered at all times, but it goes to standby once I lock my PC. Having trouble indexing and executing a query in O (log n) time.  The query includes an INNER JOIN, an ORDER BY, and an equality operation. If I understand the laws of databases correctly, a query can be indexed and executed in O (log n) time (or thereabouts) if a non-equality operator is not used on more than one column. In this case, I believe the INNER JOIN does count as an equality operator and the non-equality operator would be the ORDER BY portion of the query. This table has upwards of 10,000,000 rows and needs to handle several reads and writes per second.  Using PostgreSQL. This is what the table looks like. As you can see, the column 'Names' is a list property and it is the column that the INNER JOIN goes against:  Age Names                       Date 34  ['carla', 'john', 'sam']    3/13/2011 26  ['json', 'cindy', 'joel']   3/13/2011 72  ['beth', 'amber', 'susie']  3/13/2011 14  ['john', 'jim', 'debie']    3/13/2011   This is the query that we are trying to do:  SELECT * FROM the_table WHERE Age==26 AND Names=='john' ORDER BY Date   My background is from using App Engine's Big Table, so I've used equality operators here to indicate that 'john' should be one of the names in the Names column. This would be an acceptable query in GAE's big table, it would execute in O (log N) time as all Big Table queries are reqyured to do. I am assuming there is a way to do this in PostgreSQL as well since PostgreSQL accepts list data types as columns.  Is this possible to do in PostgreSQL?  If so, how should the index be set up (we can't figure out how to set up an index that takes into account the three properties)? 'find' utility does not output all files when using wildcards Waiting for user input with a timeout I have searched but apparently my google foo is weak. What I need is a way to prompt for user input in the console and have the request time out after a period of time and continue executing the script if no input comes in. As near as I can tell, Read-Host does not provide this functionality. Neither does $host.UI.PromptForChoice() nor does $host.UI.RawUI.ReadKey(). Thanks in advance for any pointers.  EDIT: Much thanks to Lars Truijens for finding the answer. I have taken the code that he pointed out and encapsulated it into a function. Note that the way that I have implemented it means there could be up to one second of delay between when the user hits a key and when script execution continues.  function Pause-Host {     param(             $Delay = 1          )     $counter = 0;     While(!$host.UI.RawUI.KeyAvailable -and ($counter++ -lt $Delay))     {         [Threading.Thread]::Sleep(1000)     } } Experimental observation of matter/antimatter in the universe Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron)  How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe? 'apt-get' does not work with Proxy For the command   sudo apt-get update   I get following error:  W: Failed to fetch http://ch.archive.ubuntu.com/ubuntu/dists/maverick-updates/multiverse/binary-i386/Packages.gz   407  Proxy Authentication Required (The ISA Server requires authorization to fulfill the request. Access to the Web Proxy filter is denied. )   I am running Ubuntu 10.10 installed on Windows XP using VirtualBox. For Internet connections I am using proxy server with an authentication. I tried to use gnome-network-proxy tool to set proxy settings system-wide.   After that /etc/environment has been updated by http_proxy variable with the format http://my_proxy:port/, there were no authentication data.  I checked this with Firefox. Browser asked my for login and password and everything was working fine. It was unfortunately not the case for apt-get. I have also tried to do as described . Unfortunately it does not work.  May it be somehow related to the fact that a proxy is in a Windows domain, any ideas?  My proxy name is http-proxy. Is '-' a special character here? Do I need Intel Smart Response when installing Ubuntu? With lots of guides and setup variations all over the internet, I find it confusing which is which to really follow. Normally, in Windows 8 my setup would be install everything in the HDD and use Intel Smart Response for a speedy setup. However as I am new to Linux I really don't know what to do.  My ideal setup would be something similar to this, but I know for a fact that IRST is not available in Linux so what I intend to do is keep the OS files in the SSD instead and keep my media on the larger HDD. A step by step procedure would be extremely helpful.     Edit: I want a system running solely on Ubuntu, with the SSD to speed things up but setting it in such a way that it does not hasten its lifespan.    # /etc/fstab: static file system information.  # # Use 'blkid' to print the universally unique identifier for a  # device; this may be used with UUID= as a more robust way to name devices  # that works even if disks are added and removed. See fstab(5).  #  # &lt;file system&gt; &lt;mount point&gt; &lt;type&gt; &lt;options&gt; &lt;dump&gt; &lt;pass&gt;  # / was on /dev/sda1 during installation  UUID=80289460-5983-4349-8afc-6f3119938ee9 / ext4 errors=remount-ro 0 1   # /files was on /dev/sdb5 during installation  UUID=c1678c88-5100-41bc-8090-7887f1622c7f /files ext4 defaults 0 2   # swap was on /dev/sdb6 during installation  UUID=608e3ec2-c76e-4657-af6d-80793b5f50f6 none swap sw 0 0 Been looking to pivot rows (progress_check) into columns check 1, check 2 etc... No sum or totals required, just to display the results hopefully... Can anyone help, Thanks Ad I'm trying to approximate the computer's write speed using dd:  dd if=/dev/urandom of=/dev/null bs=1K count=10000   which gives the following output  10000+0 records in 10000+0 records out 10240000 bytes (10 MB) copied, 0.746313 s, 13.7 MB/s   How can I get the '13.7 MB/s' into a bash variable? I've tried piping the output from dd to progs like awk, sed and grep to no avail.  Ultimately, I'm calling this via os.system(...) from within a python script. If anyone knows of a more direct way to get a similar result inside python I'd be interested in that also. I'm trying to predict how long a file copy will take, based on the file size.  Any help would be greatly appreciated. What are the most downvoted questions and answers on Stack Overflow, Server Fault, Meta, and Super User history to date?  Similarly, what are the most upvoted questions and answers on SO, SF, Meta, and SU to date?