I've been trying to prove this statement by induction; however, in following the steps I normally take I end up utterly stuck. I know that I must be missing something, but I have been stuck on this for a while now and can't seem to figure it out. Does anyone know how I can get to a sum on the RHS of my proof statements? Any other advice on how to find this?  Prove $**2^n &gt; n^2**$ by induction using a basis > 4: Basis: $**n = 5** **2^n &gt; n^2**$                  32 > 25 Assume: $**2^n &gt; n^2**$  Prove:  $**2^(n+1) &gt; (n+1) ^2**$  $**2^{n+1} = 2^n * 2** $    LHS Prove = LHS Assume + Fix  $**2^{n+1} &gt; (n^2) *2**  $   Utilize assumption $**2^{n+1} &gt; 2(n^2)** $      Could utilize assumption again...  $**2^{n+1} &gt; 1(n^2)**  $       ..doesn't seem to help $**2^{n+1} &gt; (n * 1)^2**   $     Basically just$ n^2$
How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct?
Is there a way to have coloured hyperref hyperlinks in the PDF, but have them print "as if" they were black? I like having my hyperref hyperlinks a different colour, so it's obvious that they are links. But when I print a copy of the PDF, I don't want the links to be coloured, because you can't click them, so it doesn't matter and it's distracting.  Other than just having a separate "for printing" version of the PDF where I turn off the hyperlinks, is there a way to have the printed version come out without the coloured links? How do I fill a shape with text, like a circle? How do I confine text within a shape such as a circle or triangle?  I don't want text along the perimeter, I would like it to break appropriately at all the inner edges. Merging two incoming broadband lines for faster Internet Which is higher — "hyper-", "ultra-" or "super-"? How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?  I ran into this identity on . I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure. We can say ‘fried fish’, ‘baked potato’ or ‘minced pork’ using past participles for modifiers.  However, ‘roast’ is different - either ‘roast duck’ or ‘roasted duck’ works, it seems to me.  How should we analyze this?  Is ‘roast’ a noun modifying a noun here, or is it a different form of the past participle?  (According to freedictionary.com, in Middle English the past participle was ‘roste’ – does that mean that this was once a strong verb?) Heat production of magnifying glass Why Magnifying glass(convex lens) produces heat when placed in front of sun? I just can't find a decent (and free) one. What can I use? Interpret geometrically the equation The function $f$ is convex when:  $$f(tx+ (1-t)x', ty + (1-t)y') ≤  tf(x,y) + (1-t)f(x', y')$$  for all numbers $0≤t≤1$ and all pairs of points $(x,y)$ and $(x',y')$ in the domain of $f$ (the whole of $R^2$)  How would I interpret this geometrically? Also what is an example of a convex function whose graph is not a plane? Travelling as an unmarried Western couple in Indonesia I keep getting asked to do wedding photography - what is a fair price to ask? This may sound like a silly post...but, I need some "outside" opinion from people who know what they are doing.  I took up amateur photography as a hobby a couple of years ago...most of it being portrait photography of my children.  As I have posted my pictures on facebook, a plethora of people have loved my "work" and have been contacting me about my "photography" business.  I have been very up-front and honest with every one of them about only being an amateur, and never professionally trained.  That being said, people still hound me to take their pictures.  I am comfortable with my ability to take portrait style pictures....and I have even started charging $60 for a two hour shoot with the rights to the cd of prints.  I do not do my own printing.    Now, I have done three weddings for family members just over this last summer....and only as a wedding gift to them (in other words, I have not charged.)  But now, I am having others, who are not family members, wanting to "hire" me for their weddings.  Again....I have tried to tell them I don't do professional photography, only to be met with all of them still wanting to hire me.    My question is...what is a fair price to ask of these people?  They obviously still want me to work for them, knowing that I have no credentials.....but I have no idea how or what to charge for.  Can anyone give me some ideas? Rotation Path Not Shown in Graph Editor I created a couple of LocRot keyframes but the rotation path and keyframes aren't shown in the graph editor (no, they aren't hidden). Which is correct: "what if there was" or "what if there were"? Possible Duplicate:          Is this correct grammar?     What if there was a Stack Overflow on…    Shouldn't it be "what if there were a Stack Overflow on…"? Encode URL in JavaScript? How do you safely encode a URL using JavaScript such that it can be put into a GET string?  var myUrl = "http://example.com/index.html?param=1&amp;anotherParam=2"; var myOtherUrl = "http://example.com/index.html?url=" + myUrl;   I assume that you need to encode the myUrl variable on that second line? Resources for Blender Since asking for tutorials, videos, resources, links to other content or downloads is considered off topic here, this is a specifically created Community Wiki which gathers resources for Blender and it has been approved by the .  Just write in the appropriate answer/section. If you have concerns, questions, post a  question, so we don't clutter the comments, but you can link your meta question from the comments.  Questions regarding such resources are not allowed anymore, except for very specific and on topic requests (ask on  if you're unsure about your question). Follow the instructions made in the question about how to post, what can be posted, etc.  Learning             Assets    (also see: )   Reference Images/Blueprints (also see: )    Tools    (also see: )      Other   Special Interest ()     Also see the . What is a word for someone without allegiance? What's the difference between an argument and a parameter? When verbally talking about methods, I'm never sure whether to use the word argument or parameter or something else. Either way the other people know what I mean, but what's correct, and what's the history of the terms?  I'm a C# programmer, but I also wonder whether people use different terms in different languages.  For the record I'm self-taught without a background in Computer Science. (Please don't tell me to read  because I'm asking this for the benefit of other people who don't already have a copy of 's marvellous book.)  Summary  The general consensus seems to be that it's OK to use these terms interchangeably in a team environment. Except perhaps when you're defining the precise terminology; then you can also use "formal argument/parameter" and "actual argument/parameter" to disambiguate. How would I write the equivalent of C#'s  in JavaScript?  var haystack = 'hello world'; var needle = 'he';  haystack.startsWith(needle) == true   Note: This is an old question, and as pointed out in the comments ECMAScript 2015 (ES6) introduced the  method. However, at the time of writing this update (2015) . Assuming a simple grep such as:  $ psa aux | grep someApp 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   This provides much information, but as the first line of the ps command is missing there is no context for the info. I would prefer that the first line of ps be shown as well:  $ psa aux | someMagic someApp USER       PID %CPU %MEM    VSZ   RSS TTY      STAT START   TIME COMMAND 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   Of course, I could add a regex to grep specifically for ps:  $ ps aux | grep -E "COMMAND|someApp"   However, I would prefer a more general solution as there are other cases in which I would like to have the first line as well.  Seems like this would be a good use case for . Understanding shape and calculation of confidence bands in linear regression I am trying to understand the origin of the curved shaped of confidence bands associated with an OLS linear regression and how it relates to the confidence intervals of the regression parameters (slope and intercept), for example (using R):  require(visreg) fit &lt;- lm(Ozone ~ Solar.R,data=airquality) visreg(fit)     It appears that the band is related to the limits of the lines calculated with the 2.5% intercept, and the 97.5% slope, as well as with the 97.5% intercept, and the 2.5% slope (although not quite):  xnew &lt;- seq(0,400) int &lt;- confint(fit) lines(xnew, (int[1,2]+int[2,1]*xnew)) lines(xnew, (int[1,1]+int[2,2]*xnew))     What I don't understand are two things:   What about the combination of 2.5% slope &amp; 2.5% intercept as well as 97.5% slope and 97.5% intercept? These give lines that are clearly outside the band plotted above. Maybe I don't understand the meaning of a confidence interval, but if in 95%  of the cases my estimates are within the confidence interval, these seem like a possible outcome? What determines the minimum distance between the upper and lower limit (i.e. close to the point where the two lines added above intercept)?    I guess both questions arise because I don't know/understand how these bands are actually calculated.   How can I calculate the upper and lower limits using the confidence intervals of the regression parameters (without relying on predict() or a similar function, i.e. by hand)? I tried to decipher the predict.lm function in R, but the coding is beyond me. I'd appreciate any pointers towards relevant literature or explanations suitable for stats beginners.  Thanks.