usually the tasks look like  $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$  or  $$\sum_{i=0}^n i^2 = i_1^2 + i_2^2 + i_3^2+...+i_n^2$$  But for the following task I have this form:  $$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^nk^3 $$  First I am a little confused by how I approach this. Do I transform them into a complete term like in the first example? Or can I do it by just using the sums themselves? And how should I treat the square of the sum best?  The first step is pretty straight forward creating the base case. But as soon as I get into doing the "Induction Step" I am getting stuck and not sure how to proceed.  I am looking to know best practice for this case.  Edit: This question is a little different, since it is expected to proove this only by using complete induction using the sum notation.
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
What are the differences between a HashMap and a Hashtable in Java? What are the differences between a  and a  in Java?  Which is more efficient for non-threaded applications? Prefix or adjective meaning "one and a half" Is there a prefix or adjective that means "one and a half", as "tri-" or "triple" is for "three"?  The exact usage I have is to describe "18" in terms of a dozen. Where I live they've started making 18-egg cartons, and the local grocer was wondering what name to give them in his computerised cash register, hoping for a single word.  Even slang, informal or invented term would do if there isn't such a term already. What's the difference when using GET or POST method? Which one is more secure? What are (dis)advantages of each of them?  () Working with a huge file. Illustrator is super slow. What can I do to ease the load? Many-worlds: how often is the split how many are the universes? (And how do you model this mathematically.) Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then must there exist a homeomorphism from $A$ to $B$ ? How to prove $n^5 - n$ is divisible by $30$ without reduction Hypernym for "audiobook", "podcast", "speech", etc.? I am looking for a short and concise term (preferably one word) that can be used as a hypernym for essentially everything that contains (audio) speech (e.g. audiobooks, podcasts, recorded news bulletins, recorded performances of comedians,..).  Does such a term exist? I've thought about 'speech', but that seems to be more adequate for e.g. political speeches, and not so much for audio books and the like. But maybe I'm wrong. How can I determine how much bleed to use? Login to Stack Exchange account doesn't use HTTPS/SSL? I know it is defined as $ dP \wedge dx  $ or explicitly,  $$ \frac{\partial P }{\partial y } dy \wedge dx   . $$  The question is, could it be $dx \wedge d P $?  Or  $$  \frac{\partial P }{\partial y }  dx \wedge dy ?    $$  I know the question must be very naive or even stupid. But I am indeed confused. How to calulate mosfet response time with load? I would like to calculate the response time of a PMOS circuit and of the circuit  Could you tell me how to do it?    I'm using this transitor.   I tried to use the rise time of the PMOS to find it can switch (and i don't know if that's the good way), but I don't now the results with the load. CL en RL. I assume CL at t=0ms is equal to 0V. I tried to use dv/dt, but I failed VIN1 = 28V. VCMD is a command  Thanks.  edit: I made a mistake. The Zener is also connected to the source. Sorry If $a_1, a_2, ..., a_n$ is a reduced residue system $\mod p$ and $a_1', a_2', ..., a_n'$ is another reduced residue system $\mod p$ then $a_1 a_1', a_2 a_2', ..., a_n a_n'$  is not a reduced modulo system $\mod p$.  Not able to show a counter to it. Help Needed. I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. This is relating to the movie. Frodo finally relinquishes his desire for the ring, and it melts in the fires of Mount Doom. The next scene shows the eruption of Mount Doom and the destruction of Barad Dur. Gandalf looks up with tears in his eyes.  I have read the books, and can't seem to remember this being explained or described. I'm just wondering why they depicted Gandalf as being so emotional about the event, when throughout the story it depicts Gandalf as having unparalleled wisdom, faith and knowledge. Out of all the Ainur he listened to Eru's song the longest and contemplated it's meaning.   It strikes me that Tolkien portrayed Gandalf as all knowing, and has ultimate faith that Frodo would carry out the task at hand. So why was he so emotionally surprised that it happened? Especially when he had seen far more surprising things happening throughout the War of the Ring and in the movies he was depicted as emotionally immovable, an almost cold hearted awareness of the evil that lurks within the world.   I understand this is probably going to be down-voted as it has alot of opinion based statements. The TL:DR is: Was Gandalf described in the book as being so emotional at Sauron's fall? Or was this simply for effect in the screenplay? What is the distinction between a professional and amateur photographer? SharePoint search not showing any results when searching partial name such as "frank" I have search service application configured.  Suppose my first name is "frankmartin" (no spaces)  When I search "frankmartin" then it displays search results in both "People" and "Site" search. But when I search "frank" it doesn't display any search result. Why is that?  EDIT  I am not using advanced search and have made this search results page using those search web parts which you can add in page to display results. 1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why? Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set.   (Can we use definition of compact set for this problem? ) I have written an essay last year, as a personal exercise. I now have a course at uni on the topic and could basically just hand in this essay now, the topic fits perfectly. I have also never submitted this essay to any other course or so.  Could I just hand in this essay, or would I violate some rules with that? Is this maybe already self-plagiarism?