Induction proof for a summation: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$ Prove by induction: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$. Hint: Use $k(k+1)^2 = 2(k+1)\sum i$.  Basis: $n = 1$ $\sum_{i=1}^1 i^3 = \left[\sum_{i=1}^1 i\right]^2 \to 1^3 = 1^2 \to 1 = 1$.  Hypothesis: Assume true for all $n \le k$.  So far I have the following:  $$\sum_{i=1}^{k+1} i^3 = (k+1)^3 + \sum_{i=1}^k i^3$$  $$(k+1)^3 + \left[\sum_{i=1}^k i\right]^2$$
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that     $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$    for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.  Thanks
Is there a word to describe someone who tends to disagree with others only to upset them? To be truly standards-compliant, must all functions in C (except for main) have a prototype, even if they are only used after their definition in the same translation unit? How do I get and modify the source code of packages installed through apt-get? How to delete all music on my iPhone How can I charge my iPhone without waking up my MacBook Pro? Applications Not Reopening After Restart Why do RAW images look worse than JPEGs in editing programs? I've found that when you load a RAW image into an editing program such as Lightroom/Aperture, the image is usually worse than if you just took the image as JPEG. Now I understand that the camera does some magic during the JPEG conversion. But I'm trying to understand what that "magic" is.   If I'm looking to implement that "magic" myself on the desktop, what kind of settings should I be trying? I find that the RAW files have more extreme contrast. The dark areas for example are way darker than JPEG. Why is this? And what is the best way to fix this? Centre of Lie Algebra $sl_2(\mathbb{F})$ For $L=sl_2(\mathbb{F})$ i.e. matrices with trace zero,  what is the centre i.e. $Z(L)$= {$x\in L : [x,y]=[y,x] \ \forall\ y \in L$}.  I will have to find matrices $A \in L$ such that $AB=BA$ for all $B \in L$. But how to appoach this considering trace is zero here. How can we extract specific files from a large tar.gz file? I found  but, when I tried the mentioned command there, I got the error:  $ tar --extract --file={test.tar.gz} {extract11} tar: {test.tar.gz}: Cannot open: No such file or directory tar: Error is not recoverable: exiting now   How do I then extract a file from tar.gz? Which tools and technologies are used to build the Stack Exchange Network? Ubuntu 18.04 opening system settings logs me out What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? Correctness of questions without inversion, relying on intonation I know that nobody likes duplicate questions, but I haven't found my answer on any of the other questions with this topic.  I built a computer with a  (made by Gigabyte).  It supposedly has 512 CUDA cores.  I followed  tutorial, (which enabled the Nvidia proprietary drivers) and was able to enable dual monitors and all the desktop graphics that are needed.  Unfortunetly, the GPU processing option still doesn't show up in Blender.  I've tried installing nvidia-cuda-toolkit, and that didn't make a difference.  I also tried to update my drivers from nvidia-346 to nvidia-349, and this had no apparent effect either.  What can I do to enable GPU rendering?  (I've tried both blender installed and just downloaded from the site). Why do you need 2 resistors when hooking up a transistor as a switch What is the point of R2 in the following diagram:   I get that R1 controls the current to the Base, but what does R2 do? gpgkeys: key C927EBE00F1B0520 not found on keyserver Possible Duplicate:    I am using 10.04 LTS - Lucid Lynx. I shutdown the computer while a software update was running. Then after I restarted my computer and while the update manager was running, I got this error:  Could not download all repository indexes The repository may no longer be available or could not be contacted because of network problems. If available an older version of the failed index will be used. Otherwise the repository will be ignored. Check your network connection and ensure the repository address in the preferences is correct. GPG error: http://toolbelt.herokuapp.com ./ Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY C927EBE00F1B0520 Failed to fetch  http://ppa.launchpad.net/rabbitvcs/ppa/ubuntu/dists/**DISTRIBUTION**/main/binary-i386/Packages.gz 404  Not Found   I went through the answers to  and I tried this $ sudo launchpad-getkeys  Please wait... launchpad-getkeys is running an update so it can detect the missing GPG keys Trying to import all the missing keys gpg: requesting key 0F1B0520 from hkp server keyserver.ubuntu.com gpgkeys: key C927EBE00F1B0520 not found on keyserver gpg: no valid OpenPGP data found. gpg: Total number processed: 0 launchpad-getkeys has finished importing all missing GPG keys.  Try running sudo apt-get update - you shouldn't see any key  errors anymore  But when I run sudo apt-get update again, it again shows that the key was not found on the keyserver. How to solve this problem? error: no such partition. Grub rescue How to find the square root of a permutation Observe that  $$\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} =  \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix},$$  so $\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix}$ is the square of a permutation. More generally, how do I check if a permutation is square and how can I find its root? How to parse data in JSON format? Do I have a virus on my phone? A message from an Android app says so I have just received a message from an Android app on my phone saying my phone has been infected with a virus and to click a link to fix the problem.   I'm not sure I trust this, is it possible to have a virus on Android?