How can I calculate this finite sum? Can someone help me?       $$\sum_{j=0}^{n-1} j^2$$
Possible Duplicate:                I know that the sum of the squares of the first n natural numbers is $\frac{n(n + 1)(2n + 1)}{6}$. I know how to prove it inductively. But how, presuming I have no idea about this formula, should I determine it? The sequence $a(n)=1^2+2^2+...+n^2$ is neither geometric nor arithmetic. The difference between the consecutive terms is 4, 9, 16 and so on, which doesn't help. Could someone please help me and explain how should I get to the well known formula assuming I didn't know it and was on some desert island?
To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. I have a question and got strucked on this.. To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. Please guide me to solve this. Thanks in advance. Rendering 2D sprites in a 3D world? How to Screen Sharing / VNC to an iMac without waking the display? I have an iMac mid 2011 at home. When I was not home, I may need to access my mac with Screen Sharing. However the iMac display will turn itself on when I connect via Screen Sharing. The problem is I don't want to scare my family at home (with the elders and kids) as the iMac "turn itself on suddenly".   This is a problem specific for iMac. On MacBooks (Pro, Air, whatever) I can set the brightness to the lowest setting so the screen is black (i.e. the LED light is off). However on an iMac you cannot turn off the LED backlit by setting brightness. The display is still bright and everything is clearly visible even on lowest brightness setting.  Therefore these solution are not working as the display will turn on as soon as you move your mouse or hit any key on the keyboard via Screen Sharing.   Sleep Display (by Ctrl+Shift+Eject) Using the code listed on    Question: Is the a way to keep the iMac display off entirely while you accessing it with Screen Sharing? Why can a function modify some arguments as perceived by the caller, but not others? $$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm  dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$  I have been trying to solve for over a week now can someone please help me. Is it bad to marinate meat for too long? From my experience it is always best to marinate meat for a decent amount of time, so the meat can "soak" up the marinade and be more flavourful.  However, I came across this recipe here:   In the directions, it says: Add pork and marinade for a minimum of 6 hours or overnight, but no longer than 12 hours.  I was wondering, why does it say "no longer than 12 hours"? Isn't it good to let meat marinate for as long as possible?  Thanks. I am looking to prove the following statement false:  Let $k$ be a positive integer, then $k\mid n$ if and only if $k\mid n^2$.  So I am trying to find a $k$ where this does not hold but after aimlessly trying numbers I was curious if there is a methodical way to find $k$. Can “whose” refer to an inanimate object? How do I get from Terminal 1 to Terminal 5 in Fiumicino Airport? Selecting the size of a System.Drawing.Icon? Capture stdout and stderr into different variables Division rules for other number systems? How could we make the same division rules for other number systems, like in our decimal system: a number is divisible with 2 if it's last digit is 0,2,4,6,8, by 3 if the sum of digits is divisible with 3, and so on.  Can we do the same rules for g-based number systems too, where g is not 10, and greater than 1? Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$. $$\lim_{x \to 0} \frac{f(x)}{|x|}=1$$ Then prove that $f$ is not differentiable at $0$. I was given this assignment and I'm not sure about my proof.. it feels like I'm assuming too much and not explaining with enough detail. Is there anything I need to add to make it sufficient? Assume toward contradiction that $f$ is differentiable at $0$. Then by definition we know that $f$ is continuous at $0$. Therefore $\lim _{x\to 0}f\left(x\right)=f\left(0\right)$ We know that $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}=1$ Therefore we can conclude $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \left|x\right|=1\cdot 0=0$ But we can also rewrite this as $\lim _{x\to 0}\left(f\left(x\right)\right)\cdot \frac{\left|x\right|}{\left|x\right|}=f\left(0\right)\cdot 1=0$ Therefore $f(0) = 0$. Since $f$ is differentiable at $0$ we have $\lim _{x\to 0}\frac{\left(f\left(x\right)-f\left(0\right)\right)}{x-0}=\lim _{x\to 0}\frac{\left(f\left(x\right)-0\right)}{x}=\lim \:_{x\to \:0}\frac{\left(f\left(x\right)\right)}{x}\cdot \frac{\left|x\right|}{\left|x\right|}=\lim \:\:_{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\frac{\left|x\right|}{x}$ Since the one sided limits of $\frac{f\left(x\right)}{\left|x\right|}$ exist, and are both equal to 1, we have $\lim \:_{x\to 0^+}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^+}\frac{\left(f\left(x\right)\right)}{x}\cdot \:\:\:\frac{x}{x}=1\cdot 1=1$ $\lim \:_{x\to 0^-}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^-}\frac{\left(f\left(x\right)\right)}{-x}\cdot \:\:\:\frac{-x}{x}=1\cdot -1=-1$ Since the limit doesn't exist, we have a contradiction to the fact $f$ is differentiable at $0$. Therefore $f$ is not differentiable at 0. $\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law] Show $R \setminus S$ is a union of prime ideals I'm stuck on the following question:     Let $R$ be a commutative ring with $1$, and $S \subseteq R$ a saturated multiplicative set (that is, $1 \in S$ and $x, y \in S$ if and only if $xy \in S$).  Show that $R \setminus S$ is a union of prime ideals.     Here's my attempt so far.  If $a \in R \setminus S$, then the ideal $Ra$ does not intersect $S$, because if $ra$ were in $S$ for some $r \in R$, then $a$ would have to be in $S$.  Thus, it follows by Zorn's lemma that there exist ideals $I$ of $R$ which are maximal with respect to the property that $I \cap S = \emptyset$.  Let $I$ be any such ideal.  If I can show that $I$ is prime, the conclusion will follow.  Let $xy \in I$.  I want to show that $x$ or $y$ is in $I$.  Suppose $x \not\in I$.  Then $I + Rx$ intersects $S$, so there exists $s \in S, a \in I, r \in R$ such that $s = a + rx$.  Then $sy = ay + rxy \in I$, so $sy$, and hence $y$, cannot be in $S$.  Similarly if $y$ is not in $I$, then $x \not\in S$.  So we are reduced to proving the following is impossible: $xy \in I$, $x$ and $y$ are not in $S$, but they aren't in $I$ either. Is there a way to set an "Adventure Mode Boundary" in Minecraft? Is there a way to set an "Adventure Mode Boundary" in Minecraft?  I am trying to create this village where players can spawn in, and then set out to explore the world and build and do normal minecraft stuff. I don't want them to destroy this village, so I want them to be in adventure mode when they are in the village and then switch them to survival when they leave, and vice-versa. When they return to the village, they get switched back to adventure mode.  In other words, I want to set an "Adventure Mode Boundary" to protect buildings. I am working without mods. I am hoping there is an simple way using command blocks. Why are the moderators being so strict with quality related flags recently? How to put a framed box around text + math environment Execute an installed Python package as a script? Lufthansa airline offers the &quot;Economy Flex&quot; fare. The listed advantages of this fare are:  Rebooking possible plus fare difference Refundable except 70 EUR plus fare difference  What does the &quot;plus fare difference&quot; mean? Imagine I buy a ticket for 200 €. Later I decide to cancel. They keep 70 Euro, so I get 130€ back. What's with the fare difference? Fare difference to what? As a comparison, the &quot;Economy classic&quot; states:  Rebooking: 70 EUR plus fare difference Refund not possible