Show that: $\binom{n}{3}= \binom{2}{2} + \binom{3}{2} + \binom{4}{2} + \binom{5}{2}+ \ldots + \binom{n-1}{2}$ This question is from Book of Proof by Richard H. Hammack. This is exercise number 13 of chapter 3 section 6. This is exercise I think is related to Pascal identity, but I'm struggling to understand it.
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$ I need a hand in showing that $$ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$$   Thanks in advance for any help.
How to construct a relative path in Java from two absolute paths (or URLs)? Given two absolute paths, e.g.  /var/data/stuff/xyz.dat /var/data   How can one create a relative path that uses the second path as its base? In the example above, the result should be: ./stuff/xyz.dat Divisibility by 7 What is the fastest known way for testing divisibility by 7? Of course I can write the decimal expansion of a number and calculate it modulo 7, but that doesn't give a nice pattern to memorize because 3 is a primitive root. I'm looking for alternative ways that can help you decide when a number is divisible by 7 by hand.  I'm sorry if this is a duplicate question, but I didn't find anything similar on the site. I want to express $$\sum_{n=-\infty}^\infty \dfrac{1}{(z+n)^2+a^2}$$ in closed form. What comes to mind is the formula $$\pi\cot\pi z = \dfrac{1}{z}+\sum_{n\ne 0}\left(\dfrac{1}{z-n}+\dfrac1n\right)=\dfrac{1}{z}+\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}$$  and also $$\dfrac{\pi^2}{\sin^2\pi z}=\sum_{n=-\infty}^\infty \dfrac{1}{(z-n)^2}.$$ But neither of these gives the term $(z+n)^2+a^2$ that we want. Perhaps we can adjust somehow? When will my device get the Android 5.0 update (Lollipop)? Android 5.0 (Lollipop) was officially announced on October 15th, 2014.  You can review the high-level changes at the  and the announcement on the official .  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 5.0, either officially or unofficially, if ever?  See also: How do I erase/format a harddisk from a live CD? I need to format a drive containing an old install of Ubuntu 10.04, to install Vista (not my idea!) Vista setup apparently refuses to format non-FAT32 or NTFS partitions. However, I can't seem to format it from my live CD (Ubuntu 11.04) at all; I keep getting "device is busy" errors. What is the simplest way to format this drive? Can I play a save from the non-GOTY edition of Red Dead Redemption in the GOTY edition? My friend is able to lend me Red Dead Redemption. However it's the vanilla edition, so I'm going to end up buying the GOTY edition anyway. Here is a question now: If I were to play Red Dead Redemption, non-GOTY edition, have a save from it, and then buy the GOTY edition, will I be able to play on the save I made while playing the non-GOTY edition? Who were the hooded people in The Rise of Skywalker? Near the end of Star Wars Episode IX: The Rise of Skywalker there are several hooded figures that are shown chanting stuff (in Sith?) and just hanging out with Palpatine. Who were these people? How do I programmatically log in a user with a POST request? Problem with Refsection How to participate in clan war instead of being a spectator I joined a new clan and now that the clan war has started I tried to attack but it says "Spectator can't attack". How do I get out of being a spectator (I haven't used either of my 2 attacks in the war). I went to the clan castle and the clan option and I selected "Yes I'll participate in this war" but nothing happens. 1970s scifi/horror novel where protagonist is used by a crablike creature to feed its larvae, goes mad, and is defeated by retraumatising him I read this book once as a teenager, I think it might have been written in the 1970s but could be late 1960s or early 1980s.  The protagonist is left to die hideously by enemies. He is paralysed and left on some planet (I don't believe it was earth) in the habitat of some kind of crablike creature, which injects him with its young, and periodically returns to force nutrient and a paralytic agent down his throat, as the larvae grows inside him.  He is rescued, and his body rebuilt to some kind of superhuman abilities, but he is traumatised and bent on indiscriminate revenge for his suffering. He proves hard to defeat.   He is eventually defeated by trickery, when he is shown (or demands to see) something by some kind of video link or remote TV/monitor - perhaps he is told that people he wants revenge on, are held captive for him, I can't be sure.   I think maybe he takes hostages or makes a severe threat, and his demands relate to the people he wants revenge on. But he can't be physically reached, so what can be done?  They set up a video feed, to negotiate, or more likely, to purportedly show him they have what (or who) he wants, and he looks to see.  When he looks at the video stream, it's actually an image of the crablike creature. The shock of repressed memories flooding back, renders him re-traumatised and perhaps catatonic, and defeats him. English (other than American English) has a clear differentiation between the two words. Both are about translocating something. In "bring" the something of somebody is moved to where the speaker is currently situated. "Take" is used to translocate something or somebody to a place that the speaker is not currently at.   You cannot “bring” your books to school if you are currently at home. You can only take them to school. Most of the time one can get the meaning from the context of the sentence but it can get very confusing when the other party is on the telephone and you do not know their location.   Why does American English not differentiate and when did it lose the differentiation? Windows 7 booting fail with error code 0xc0000225 I have an HP laptop Pavilion DV4. Last night, I installed Ubuntu and there was an problem about partitioning the hard disk, and I think I deleted the Windows boot record.  I didn't burn a recovery disc but there is built-in recovery partition on the hard disk.  All I want to do is to perform a Factory Restore but I need recovery disc. I could burn that disc from the recovery partition using HP recovery manager, but can't burn now. I can't also boot HP recovery manager using F11 key, because it has the same boot error.  I want to boot Windows 7 and want to burn recovery disc. How can I boot into my old windows 7. If I can boot it, I can burn Recovery Disc and I will make Factory Restore if it's needed.  I didn't make any changes to windows 7 partition but I did a problem about Boot Record.  There is a system partition on hard disk with the size of about 200 MB.There is BOOT folder too.  Installation of Ubuntu also failed, so I'm booting from Live Ubuntu USB. How do I access a page's HTTP response headers via JavaScript?  Related to , which was modified to ask about accessing two specific HTTP headers.     Related: Quoting in ssh $host $FOO and ssh $host "sudo su user -c $FOO" type constructs I often end up issuing complex commands over ssh; these commands involve piping to awk or perl one-lines, and as a result contain single quotes and $'s.  I have neither been able to figure out a hard and fast rule to do the quoting properly, nor found a good reference for it.  For instance, consider the following:  # what I'd run locally: CMD='pgrep -fl java | grep -i datanode | awk '{print $1}' # this works with ssh $host "$CMD": CMD='pgrep -fl java | grep -i datanode | awk '"'"'{print $1}'"'"   (Note the extra quotes in the awk statement.)  But how do I get this to work with, e.g. ssh $host "sudo su user -c '$CMD'"?  Is there a general recipe for managing quotes in such scenarios?.. How can I make a single event handler handle ALL Button.Click events? How can I let player click text to use a command? I want players to be able to click text to run an OP command even if that player is not an OP. For example something like:  Console: Do you want to enable Fly? [If you do, Please click here]  *Player1 clicks that text*  Console: /fly Player1   Is something like this possible? Can't delete folder in Windows 7 I'm trying to delete a folder in Windows 7 and get a perplexing error message: "Could not find this item:  This is no longer located in G:\Graphics.  Verify the item's location and try again.  I can see the folder -- I can find it.  I just can't delete it.    I also get a second error message (sometimes) when I click on the folder: G:\Graphics 2009-11-17 refers to a location that is unavailable...this information might have been moved to a different location.    I'm using Windows 7; this folder is on an external hard drive.  I've emptied the folder (there were items in it); I've scanned that external hard drive for errors.  Trying to rename the folder yields the same enigmatic error message.    Is there a way to delete this folder? We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit. Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above?