Simple proof by induction: $1^3 + 2^3 +3^3 +...+ n^3 = [ (n(n+1))/2 ]^2 $ I am rather illiterate when it comes to mathematics, I am afraid. In an effort to change that, I grabbed a copy of 'What is mathematics? : An elementary approach to ideas and methods' and have already encountered some difficulty. It seems silly to waste so much time trying to solve it myself, so I decided to ask for some help in solving and illuminating the taken steps, so that I can solve some more on my own. I shall review some of the basics while awaiting answers, and hope that my session tomorrow shall be more productive...  The problem is as follows:  Prove by induction that $1^3 + 2^3 +3^3 ... n^3 = [ (n(n+1))/2 ]^2 $.  As is, at the point I decided to seek help and look up material for review, I have taken the following steps:  solved for the base case: n=1, $1^3 = [2/2]^2 $  $$ 1 = 1^2, 1=1 $$  Then after proving the basis, I stated the assumption that:  $1^3 + 2^3 ... k^3 = [ (k(k+1)) / 2 ]^2$ is true.  Then I tried to solve for the next case, $(k+1)^3$:    $$ 1^3 + 2^3 + 3^3 ... k^3 + (k+1)^3 = [ ( (k(k+1)) / 2 ) + (k+1) ]^2 $$  $$ [ (k(k+1)) / 2 ]^2  + (k+1)^3   =  [ (k(k+1) + 1(k+1) ) / 2 ]^2 $$  at this point after failing for a while and having spent quite some time looking up material, I sought help. I hope I'm going down the right path here, but I suspect I shall soon find out...   Help is greatly appreciated.
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
[Firmware Bug]: TSC_DEADLINE disabled due to Errata - what should I do about this? Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these  and saw the quaternion group, but the subgroups of the quaternion group aren't abelian... Allow questions to be saved as drafts prior to posting Android: Making Https Request I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius.  I have managed to solve find the constants for $||.||_{1}$ and  $||.||_{2}$ but I cannot see how to continue if I want to prove the following:  $||A||_{\infty} \leq \sqrt{n}||A||_{2}$   and the ones for Frobenius:  $||A||_{F} \leq \sqrt{n}||A||_{1}$  $||A||_{F} \leq \sqrt{n} ||A||_{2}$  Any help is appreciated. Thank you for your time "a ten years old boy" or "a ten year old boy" 1) a ten years old boy is sitting on the couch.      2) a ten year old boy is sitting on the couch.      3) a ten-years old boy is sitting on the couch.      4) a ten-year old boy is sitting on the couch.   Which is correct? I generally know how to use a hyphen, but when should I use an en-dash (–) instead of an em-dash, or when should I use a hyphen (-) instead of an em-dash (—)? Can I do it with reflection or something like that? Computing the $n^{\textrm{th}}$ permutation of bits. I've seen  post about the $n^{\textrm{th}}$ permutation of a set but that is not what I need. If you have a bit string (ones and zeros only) there are  to quickly permute the NEXT lexicographically ordered bit permutation. For example take $$000111 \rightarrow 001011 \rightarrow 001101\rightarrow\cdots$$ etc. If the string is long then there are going to be approximately one Bajillion of these things. And I want to know how to compute the $n^{\textrm{th}}$ guy without doing the exhaust.  Backgound: This is for a parallel computing job where I need to farm out the search space of elements of a set.  Foreground: I own a copy of Knuth's 4th volume of "The Art of Computer Programming" and I think the answer is in there but I can't seem to find it. (It's like 900 pages).  I'm posting here in the hopes that someone has knowledge of this (obviously). Even if you can point me to a source, say the part in Knuth's book where he describes this problem I would be most grateful. Which languages was Gandalf proficient in? A word for a worldly wise person who pretends to be naïve? What is a word for someone who is experienced and wise, but who deliberately acts naïve?    I don’t intend it for sarcastic use; I’m trying to describe someone succeeding in making people think that that person truly is naïve.   Is there a word for this? I've read the , and I don't get it. I am really interested, but I've read the explanation there a couple of times and I still don't get it how the Duff's device works.  What would a more detailed explanation be? Should 'fuck' be present in review audits (intentionally fake suggested edit reviews)? Some suggested edit reviews are fake and are intended to check that the reviewer is paying attention and reviewing correctly.  I recently encountered one where the fake suggested edit contained 'fuck':   Despite the fact that we regularly encounter profanity online these days, and whilst it is not uncommon to encounter the word 'fuck' in blog posts by recognised authorities in the world of software development (and, I assume, in other fields), this nevertheless caught me a bit by surprise.  I have no problem with whatever strength of language someone feels is appropriate and relevant to make their point, however in the context of audit-related fake suggested edits, is the use of profanity appropriate?  Update: just to clarify, the word 'fuck' was not present prior to the fake suggested edit. The fake suggested edit added the word 'fuck' where it was not previously present. I just finished a test as part of a job interview, and one question stumped me, even using Google for reference. I'd like to see what the StackOverflow crew can do with it:  The memset_16aligned function requires a 16-byte aligned pointer passed to it, or it will crash. a) How would you allocate 1024 bytes of memory, and align it to a 16 byte boundary? b) Free the memory after the memset_16aligned has executed.  {        void *mem;    void *ptr;     // answer a) here     memset_16aligned(ptr, 0, 1024);     // answer b) here     } can't login as mysql user root from normal user account in ubuntu 16.04 I just installed Ubuntu 16.04 LTS along with the packages php, mariadb and nginx. I ran mysql_secure_installation and changed the root password.  Now when I try to login to mysql using the root account while logged in Ubuntu as normal user account I get access denied.   When I login using sudo mysql, mysql doesn't even ask me password. If I run mysql_secure_installtion I see that old settings never got set permanently.   What am I doing wrong? Brackets in if condition: why am I getting syntax errors without whitespace? What is the quickest way to send some text from desktop\laptop to iPhone? I usually send emails to myself, but I wonder if there is a better way to share something quickly from my PC\laptop to iPhone. Explanation of seasons in "A Song of Ice and Fire" Google Chrome will take up my memory to the point where it causes my computer to freeze to a near halt. What can I do to prevent this? It gets so horrible that I can not even switch to TTY1 to force a reboot. I have to hard reset, and I lose work.  Ideally, I would like to be able to use the chrome browser on Ubuntu without having to keep reminding myself to close tabs.   Are there any extensions for Chrome, or tools for Ubuntu that can help prevent me going past the point where Ubuntu eats up all the RAM and becomes incapacitated? Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$.