$$\sin(\theta)+ \sin (2\theta)+\ldots+\sin(n\theta).$$  Find the identity? Set $z=e^{i\theta}$.  Can someone point me in the right direction as to where to start/go from.  So far I have  $$\frac{1-e^{i\theta(n+1)}}{1-e^{i\theta}}.$$  And I am stuck as to where to go from here.
I'm trying to solve    So far I've done the first part, evaluating the summation ;  where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I understand that the summation of simply $$\sum\limits_{k=0}^n\cos(kx)$$ is derived from looking at the real part of  $$\sum_{k=0}^n{\rm e}^{ikx} $$I'm guessing they want me to see how the second summation in the question is defined by 'playing around' with the original?
Printing the value of a variable in SQL Developer Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? Minecraft Survival Mode I am playing Minecraft on survival mode. I can break blocks like sand, dirt, and gravel and it will give me a block. When I try to break stone or iron or anything else, it didn't give me a block. What can I do so that I can get a block? I am on single player and my internet is fine. I tried deleting the app and then getting the app again but that didn't work. This is happening with all my worlds in survival mode. Proof a $(2n-1)$-compact manifold I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold.  How give the charts. For $n=1$ and $d=2$ not distinguish which 1- manifdold is? How can I make a surface look wet in cycles? I am trying to recreate a wet road with lights reflecting off of it. I started by making the object glossy but this did not give a very realistic look  I think I need to increase the specularity of the reflection to make it look more real but I can't find this option anywhere in cycles. Also what other techniques can I use to make a surface look wet in blender? I can link directly due to licening but  is what I am trying to recreate. First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity? I'm trying to generate a secret key to be used for HMAC SHA-256 signature processing. I've seen many sample of keys with variable length from 32 characters to 96 characters.  What is the ironclad rule for this key size? Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=ba_n=0$? Copying unicode symbols from Firefox address bar as is Let's say I open a webpage with some Unicode characters, say, Cyrillic, in the address like this:        When I try to copy it from the address bar somewhere else, it becomes unreadable rubbish:        I guess this is for compatibility. However for readability I want to copy it straight away with proper Unicode characters.  What and how should I tweak to make that possible?  Edit: The answers are good, but they are hacky. Are there any settings in about:config? What are the barriers to understanding pointers and what can be done to overcome them? Why are pointers such a leading factor of confusion for many new, and even old, college level students in C or C++?  Are there any tools or thought processes that helped you understand how pointers work at the variable, function, and beyond level?  What are some good practice things that can be done to bring somebody to the level of, "Ah-hah, I got it," without getting them bogged down in the overall concept? Basically, drill like scenarios. How long is it safe to marinate meat? Are there specific techniques/filters for snowscapes? Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials.  We have that $R[[x]]$ is a ring containing $R[x]$ as a subring.  How to prove that if $a_0\in R$ is a unit, then  $\displaystyle\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$? Output from ls has newlines but displays on a single line. Why? How can I get a complete list of fields for a standard object? I am new to Apex, but it is apparent to me that most (if not all) standard objects probably have an ID field. However, this field does not appear in the reference for the standard objects in the Soap API Developer's Guide. Instead, it reads:     To verify the complete list of fields for an object, you can use a   describe call from the API, or inspect with an appropriate tool, for   example, inspecting the WSDL or using a schema viewer.   If I want to find out for sure whether an object has an ID field, or what the field is named (is it ID, Identifier, id, etc), it sounds like I need to use a "describe call" or "an appropriate tool" or inspect the "WSDL" using a "schema viewer".  Now, I am an experienced software developer, but those solutions are so ambiguous I have no idea where to look next to figure out how to get an object's ID. Can anybody either refer me to more specific documentation, or explain how to find the complete field documentation via a "describe call", "appropriate tool", or "schema viewer"? What is the optimal path between $2$ fixed points around an invisible obstructing wall? Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). The wall extends $1$ mile in each direction perpendicular to the line segment (direct path) between A and B, and its position can be at any random location between the two points. That is, it can be any distance between A and B such as $0.1$ miles away from A, $1.5$ miles away, $2.9$ miles away.... You don't know if the wall is present or where it is until you walk into it.  You must walk around the wall if present as there is no other way to circumvent it.  Assume the wall is negligible thickness (like a force field), all ground is perfectly flat, and the y coordinates at both A and B are $0$ (although I don't think the optimal path will change much if they weren't).  What strategy minimizes the average expected walk distance between A and B?  How do we know for certain this strategy yields the shortest average walking distance?  To get the $100$ bounty points, I would like a convincing argument from you as to why you feel your solution is optimal and cannot be beaten.  For those of you using computer simulation, anything reasonably close to optimal is a candidate for the bounty.   Can anyone beat the optimal solutions submitted so far?  There are still a few days to try.  Even a slight beat (if not due to roundoff error or some other error) would help prove previous solutions are non-optimal.  You can just use the table of $31$ points (x-y coordinates) and compute that path length and then if you can find better then I will accept that answer.  I think by Sunday night I may have to award the bounty otherwise it may get auto-awarded. What's the intuition behind difference of $\neg(S\land E)$ and $\neg S \land \neg E$? Let $S$ represent the English statement "Sales will go up."And let $E$ represent the English statement "Expenses will go up." Now my problem is that I have trouble telling the difference between these two statements: $\neg(S\land Q)$ and $\neg S \land \neg E$. When these two statements are translated into English they sound,to me, the same. However, when you look at their truth tables they are different.   Here is my best translation of these two statements:  1) $\neg(S\land E)$ represented in English, "Both sales and expenses will not go up."(Correct me if I translated it wrong)  2)$\neg S \land \neg E$ represented in English, "Expenses will not go up and sales will not go up."(Again, correct me if I'm wrong)  I still do not understand the difference between these two statements in the English representation. They both sound the same thing to me.  Once more, I have the same trouble I have with understanding with the or-connective( $\lor$ ). Again let E and S be the same statements as before.Tell me if I might have translated them wrong.  1) $\neg (S \lor E)$ represented in English is "Neither sales nor expenses will go up."  2)$\neg S \lor \neg E$ represented in English is "Sales will not go up, or Expenses will not go up."  Yet I still have trouble understanding the difference between the English translation.  In short, can you give me some alternative example that better explains the difference, something intuitive that gives me a clear picture of the difference of the meaning between the statements I showed with the connective-or and connective-and. Thank you. Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$ Is there a way to edit SVG files in Illustrator where you can specify a CSS class for each path?  I already know in Illustrator that if you give the layer name an actual name, Illustrator will use that name as the path's ID, which is fine if you don't plan on reusing the icon more than once on the same page.  My current workaround is to just use the IDs method, but then later convert the IDs to classes in my code editor, but it's pretty annoying to have to do every time I generate my SVG sprite.  If that's currently not possible in Illustrator, are there any other apps that will allow you to specify that? Or maybe a Grunt or Gulp package?  It looks like you might be able to do it with Inkscape with a , so I might look at that if there's no other good solutions out there. counting triangles in a graph or its complement Given a simple graph $G$ [no loops or parallel edges] on six vertices,  let $G^c$ denote its complement. It is known that either $G$ or $G^c$ must contain a triangle  $T$ in it. [An example of a Ramsey number I think.]   My question came up when I tried to find such a graph on six vertices  which had only one $T$ in it or its complement.  I believe if I did things right there are none of these. This I tried to check by drawing a copy of each graph $G$  on six vertices which has a $T$ in it,  and then looking at the complements to see if any had  no $T$'s in them. I found none.  One way to pose my question for a general number $n$ of vertices  is then as follows: Consider the set of all graphs $G$ on $n$ vertices,  and for each one construct $G^c.$  Count the number of $T$ in $G$ and add that to the number of $T$ in $G^c.$  Finally find the minimal such sum for the given $n$  and call that say $M(n).$ So my above six vertex conjecture  is that $M(6)=2.$   By considering a cycle graph on five vertices we find $M(5)=0.$   So going further one could ask if anything is known  about this minimum function $M(n)$ for larger $n$ [such as bounds, etc.] And does it have a name?  I would also like to see a simpler way to handle the six vertex case, one not involving drawing all the graphs.