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How to make the simple three square box and connect them? I want to make the three square box and simply connect them. Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$? Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated. Reading a binary file and using Response.BinaryWrite() What is average distance from center of square to some point? How to check if installing a package will require a reboot? Google Maps API map doesn't appear How to understand Cauchy's proof of AM-GM inequality(the last step) The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$  the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident.  then cauchy use this fact repeatedly. He get: $$a_1a_2\cdots a_{2^m} \leq \left(\frac{a_1+a_2+\cdots +a_{2^m}}{2^m}\right)^{2^m}.\tag{$\ast$}$$  This is easy to understand, and natural, but next step:  he let  $$b_1 = a_1,\ b_2=a_2,\ \ldots,\  b_n=a_n,\  b_{n+1}=\cdots=b_{2^m}=\frac{a_1+a_2+\cdots a_n}{n}$$  replace $a_1, a_2,\ldots ,a_{2^m}$ with $b_1, b_2, \ldots b_{2^m}$ in $(\ast)$ and simplified both left and right. He get the answer. I can repeat his proof, but I don't know why Cauchy can got this idea. How does he got it? A fair coin is tossed repeatedly until 5 consecutive heads occurs.   What is the expected number of coin tosses? We are small team which working on car racing game but we don't know about licensing process for branded cars like Nissan, Lamborghini, Chevrolet and etc.   Do we need to buy any licence for using real car brand names, models, logos,... or we can use them for free?  Second option we think about using not real brand with real models is it possible? If someone have experience with that, fell free to share it.  Any information about that is welcome. Propane gas fireplace pilot won't light I've asked this question before. Now I've posted some pictures as I can't figure out where I should clean. I don't smell propane. I tired lighting manually with a lighter and nothing. I know to turn the valve on,  rotate the wheel to pilot. Push in and try igniter. It worked fine 6 months ago.   Any suggestions? why does fdisk add one additional sector when creating partitions? I'm playing around with fdisk and I tried to create a partition on an empty USB drive. Here's how that went: Command (m for help): n Partition number (1-128, default 1): 1 First sector (34-61187102, default 2048): 2048 Last sector, +/-sectors or +/-size{K,M,G,T,P} (2048-61187102, default 61187102): +30000000           Created a new partition 1 of type 'Linux filesystem' and of size 14,3 GiB.  so just one partition with 30 million sectors. However, when I check how many sectors it contains I get:  I'm checking with a different program and I see again that additional sector:  Sorry for the pettiness :) I'm learning. How to create LV(Logical Volume) after ubuntu is installed? everyone. I need to create LV for xen guest. There is only one disk of 70G. While installing, I use the default process and not the LVM. Do I need to reinstall Ubuntu to create the LV? Is there some better method? Thanks a lot.  root@vswitch-BlackfordESB2:/home/vswitch# vgdisplay   No volume groups found root@vswitch-BlackfordESB2:/home/vswitch# vgcreate vm /dev/sda1   No physical volume label read from /dev/sda1   Can't open /dev/sda1 exclusively.  Mounted filesystem?   Unable to add physical volume '/dev/sda1' to volume group 'vm'. root@vswitch-BlackfordESB2:/home/vswitch# pvdisplay  root@vswitch-BlackfordESB2:/home/vswitch# df -h Filesystem            Size  Used Avail Use% Mounted on /dev/sda1              60G  4.2G   53G   8% / udev                  3.6G  4.0K  3.6G   1% /dev tmpfs                 1.5G  1.1M  1.5G   1% /run none                  5.0M     0  5.0M   0% /run/lock none                  3.6G     0  3.6G   0% /run/shm What license does Ubuntu fall into (GPL, MIT, a mix)? Would it be legal to modify it and redistribute my modified version? How do I reset my manor so I can rebuild it with different options? I currently have Windstad Manor and built it all. Now I regret some choices and I'd like to rethink and rebuild it.  Is there any console way to "nuke" the manor and rebuild it from ground up? No hyper refs from citation to bibliography using BiBTeX, hyperref and harvard $a^2b^2+a^2c^2+b^2c^2\leq 3$ Let $a, b, c\geq 0$ s.t.$$(a^2-a+1)(b^2-b+1)(c^2-c+1)=1$$  Show that $ a^2b^2+a^2c^2+b^2c^2\leq 3$.  My idea:  I denote $a+b+c=x$, $ab+bc+ac=y$ and $abc=z$.  Then I have $$x^2+y^2+z^2-xy-xz-yz+2z-y-x=0$$ I have to show that $$y^2-2xz\leq 3$$ I tried to prove it with the sign of trinom, but it doesn't work. I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go.  In the truth table, where $A \implies B$ we obtain this result:  A   B   T/F ___________ T | T | T T | F | F F | T | T F | F | T   Now, my confusion here is regarding why $A\implies B$ given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.  Now... then, why is B true despite the fact that A is false? Or rather, why is the statement B given not A, True. The truth of B is implied from the truth of A, but A is a false statement. So it's very contradictory that this may be, and in fact, if you go through a real life example, you would find that the third statement seems to be false.  Say $A$ is the statement that it is dark outside, and $B$ the statement that the sun is not on our side of the earth. If it is not dark outside, than stating that the sun is on the other side of the planet must be wrong (in my example i'm going to claim that the sun is the only light source).  Side note:  I just had a thought, if A is false, does that mean you can conclude... mmm, whatever? I mean, B is implied via the truth of A, if A is false, B is NOT implied, but then B could be EITHER true or false, I suppose. Is this correct?  However, would that not mean that A does not imply anything when false. Would that not set the 3rd and 4th statement to be undefined? Image of the union is the union of the images $$ f\left(\bigcup\limits_{\lambda \in \wedge} A_{\lambda}\right) = \bigcup\limits_{\lambda \in \wedge} f(A_\lambda)$$  Let $b \in f(\bigcup\limits_{\lambda \in \wedge} A_{\lambda})$ $\rightarrow b=f(a)$ for some $a \in (\bigcup\limits_{\lambda \in \wedge} A_{\lambda})$.  Since $a \in (\bigcup\limits_{\lambda \in \wedge} A_{\lambda}) \rightarrow a \in A_{\lambda}$ for some $\lambda \in \wedge$.  Since $b=f(a)$, then $b \in f(A_{\lambda})$ for some $\lambda \in \wedge$.  $\rightarrow b \in \bigcup\limits_{\lambda \in \wedge} f(A_\lambda)$ $\rightarrow f(\bigcup\limits_{\lambda \in \wedge} A_{\lambda}) \subset \bigcup\limits_{\lambda \in \wedge} f(A_\lambda)$.  I used a similar argument to prove $\bigcup\limits_{\lambda \in \wedge} f(A_\lambda) \subset f(\bigcup\limits_{\lambda \in \wedge} A_{\lambda})$.  Which shows $ f(\bigcup\limits_{\lambda \in \wedge} A_{\lambda}) = \bigcup\limits_{\lambda \in \wedge} f(A_\lambda)$.  Please let me know if this is valid. What could happen if I use 2 cycle fuel in a 4 cycle engine? First off, this is NOT to be confused with the following question:    2 cycle oil is used for my weed eater. There is no way I could use a whole gallon in a year.  Sometimes I run out of gasoline for my 4 cycle lawn mower, and that 2 cycle mix fuel is sitting there getting really old.  If I run that in the lawn mower, am I simply going to create extra pollution or could the 2 cycle fuel cause any damage to the 4 cycle engine? Can X.org Load Two Different Drivers? (Triple-Head Setup with Intel and NVidia Card) We want to set up a 3 screens with an onboard intel graphics adapter and an nvidia card.  Both cards are recognized in lspci   01:00.0 VGA compatible controller: NVIDIA Corporation GT218 [GeForce 210] (rev a2) 00:02.0 Display controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   And bascially work: If I switch to the console, I see output on the Intel card, X is only running on the Nvidia card.  But when I type xrandr | grep conn I get only one connected port:   DVI-I-0 disconnected (normal left inverted right x axis y axis) VGA-0 disconnected (normal left inverted right x axis y axis) DVI-I-1 connected 1920x1200+0+0 (normal left inverted right x axis y axis) 518mm x 324mm HDMI-0 disconnected (normal left inverted right x axis y axis)   I am on Ubuntu 12.10 (and XFCE) and installed the proprietary nvidia driver.  I was assuming that X maybe has a problem to autodetect both cards, so I thought the following xorg.conf could help:   Section "ServerLayout"     Identifier     "Layout0"     Screen      0  "Screen0"     Screen      1  "Screen1"     InputDevice    "Keyboard0" "CoreKeyboard"     InputDevice    "Mouse0" "CorePointer" EndSection  Section "Files" EndSection  Section "InputDevice"     # generated from default     Identifier     "Mouse0"     Driver         "mouse"     Option         "Protocol" "auto"     Option         "Device" "/dev/psaux"     Option         "Emulate3Buttons" "no"     Option         "ZAxisMapping" "4 5" EndSection  Section "InputDevice"     # generated from default     Identifier     "Keyboard0"     Driver         "kbd" EndSection  Section "Monitor"     Identifier     "Monitor0"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device0"     Driver         "nvidia"     VendorName     "NVIDIA Corporation"     BoardName      "GeForce 210" EndSection  Section "Screen"     Identifier     "Screen0"     Device         "Device0"     Monitor        "Monitor0"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   Section "Monitor"     Identifier     "Monitor1"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device1"     Driver         "intel"     VendorName     "Intel"     BoardName      "Internal Intel Graphics" EndSection  Section "Screen"     Identifier     "Screen1"     Device         "Device1"     Monitor        "Monitor1"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   But it is still not working.   Is this generally possible to use two different drivers in X ? Is it necessary to create an xorg.conf manually then? (because Ubuntu 12.10 normally works without the xorg.conf) Are there any errors in my xorg.conf?