I have come across a problem in a book on elementary mathematics that I am unable so solve. The problem statement is : &quot;Show that the numbers $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ cannot be terms of a single arithmetic progression&quot; When I look at the hint for the problem in the back of the book it provides a partial proof by contradiction. They say : assume we have an arithmetic progression such that $m,n,p \in \mathbb{N}$ with : \begin{align} a_{m} &amp; = \sqrt{2} \\ a_{n} &amp; = \sqrt{3} \\ a_{p} &amp; = \sqrt{5} \end{align} Here we know $m$, $n$, and $p$ must be pairwise distinct. They then tell the reader to use the formula for the general term of an arithmetic progression to compute $\frac{n-m}{p-m}$. The formula referenced here is : \begin{equation} a_{k} = a_{1} + (k-1)r \;\; \forall k \geq 1 \end{equation} Where $(a_{i})_{i \geq 1}$ is the progression and $r$ is the common difference between adjacent terms. I am able to compute $\frac{n-m}{p-m}$ using the formula, but I do not understand why computing this can allow proving what the problem requests of the reader. Can someone help me with this ?
In a sequence $a_1, a_2,\dots$ of real numbers it is observed that $a_p=\sqrt{2}$, $a_q=\sqrt{3}$, and $a_r=\sqrt{5}$, where $1\leq p&lt;q&lt;r$ are positive integers. Then $a_p$, $a_q$, $a_r$ can be terms of (A) an arithmetic progression (B) a harmonic progression (C) an arithmetic progression if and only if $p$, $q$, and $r$ are perfect squares (D) neither an arithmetic progression nor a harmonic progression  I have tried using the definition of AP and argued that if the first option is true then $\frac{\sqrt{3}-\sqrt{2}}{\sqrt{5}-\sqrt{3}}=\frac{q-p}{r-q}$  is true which implies that the left hand side is an rational number. I don't know whether the left hand side is a rational number or not. I am stuck here. Also, I don't know how correct this approach is. How do I approach and solve this problem?
What's the difference between  and ,  and ? What's the difference between &lt;b&gt; and &lt;strong&gt;, &lt;i&gt; and &lt;em&gt; in HTML/XHTML? When should you use each? Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?   I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps shouldn't change the ratio of radii between two circles.   Would it be correct to just say that if $\frac{r_1}{r_2}$ is the ratio of radii between the two circles, then   1) The radii are invariant under translation, $z \mapsto z+a$, so $\frac{r_1}{r_2}$ stays the same  2) Under scaling by a factor $z \mapsto az$, $\frac{ar_1}{ar_2} = \frac{r_1}{r_2}$   3) Under inversion, $z \mapsto \frac{1}{z}$, $\frac{1/r_1}{1/r_2} = \frac{r_2}{r_1}$  Or is there a different/better way to think about this problem? R2jags does not remove the burn in part sometimes? Apt-get 403 Forbidden, but accessible in the browser I have been stuck for several days on this old Analysis problem (I am doing some study on my own).  I have tried several things (which I'll indicate below), but I cannot seem to figure it out.  Here is how the problem is presented: Problem: &quot;Let $f$ be a continuous real-valued function on $[a,b]$.  Suppose there exists a constant $M \geq 0 $ such that $$|f(x)| \leq M \int_a^x |f(t)| dt$$ for all $x \in [a,b]$.  Show that $f(x)=0$ for all $x \in [a,b]$.&quot; My Thoughts: I have tried using the mean value theorem iteratively, but that seems to always lead me down a dead end road.  I deduced that $f(a)=0$.  If only $f$ were assumed to be differentiable, then maybe I could play with trying to get the derivative to be $0$, but unfortunately it's only continuous.  My other thought was to (somehow) use the condition to show that $\int_a^b |f(x)| dx = 0$.  I also played around a bit with contradiction, but to no avail.  Even if one of these hair-brained thoughts is correct, I am not really sure what to do next. If you have any ideas, suggestions, or solutions, I would really appreciate it if you are willing to share them.  Thank you for your time. Short story about robots on a farm after losing communications, go out to rule the world Does decrease in temperature affect mass $E=mc^2$? My understanding of Quantum physics and String Theory is very basic and I don't yet have a grasp on the maths, but in my research I have come up with a question.  Does a decrease in temperature also signify/create a decrease in mass?  I am quite willing to believe the concepts I think I am beginning to understand are totally misconceptions on my part.  But, as I see it mass is a product/mark of disruption to the Higgs field as a particle moves.  If these particles are ultimately composed of vibrating Strings, their mass is given by the rate at which they vibrate and how it interacts with the Higgs field.  I don't have any idea if, as the temperature (as we know it) approaches 0k, the string/quark/sparticle would change its state in any way or if the changing of the state of the String would change what type of particle it represents.  It seems I'm more full of questions than answers, but that's why I love these topics!  I know what I'm learning is new if the best and brightest are still working on the 'basics' to just make their systems work :). How to change the style of the title attribute inside an anchor tag? Saving Photoshop multilayer file to PDF without layers size difference Why is it that when I save a file that has many, many layers to a PDF without layers as a copy that the file size is larger than flattening the layers and then saving the file as a copy to PDF? 19Mb vs. 5Mb?? Wouldn't saving a file without layers be the same as flattening the layers and then saving the file?? Can anyone explain this? Hyphenation in compound adjectives Possible Duplicates:                In the sentence "Portland is known to be one of the most bike friendly cities in the US", is a hyphen necessary in "bike friendly"? As far as I know, the hyphen is only required when leaving it out would cause ambiguity. Wikipedia uses the example of "a small appliance factory" vs "a small-appliance factory".   As far as I can tell, the sentence above is pretty clear. A quick google search for "bike friendly city" brings up instances of both hyphenated and non-hyphenated usage. Thoughts?  Please excuse the poor formatting, sent from my iPhone. "Welcome to the Jones__": Plural or Plural Possessive? When making a welcome sign, what form of the name Jones would you use?  ("Welcome to the Jones__") How to get domain name from URL When testing the difference between 2 proportions why do we use a z-test rather than a t-test?    Further, is there a simple way to conduct an omnibus test for significant differences between more than 2 proportions (in the form of percentages). Is there an equivalent to a one-way ANOVA for this?  I imagine that you could use logistic regression (assuming you have the original data for the proportions in the form of 0s and 1s) Are there other options? Security importance of Key Schedule in Block Cipher How to identify a downvoting stalker Possible Duplicate:          Yeah I got a deranged stalker that keeps downvoting me.  Anyone know how going about identifying such individuals?  (I just got 4 all at the same time just after making a new post).      that was the question I originally asked. Please tell me, what is so wrong with it ? Can I load a 32 bit DLL into a 64 bit process on Windows? I am developing a game with a Minecraft-like terrain made out of blocks. Since basic rendering and chunk loading is done now, I want to implement block selecting.  Therefore I need to find out what block the first person camera is facing. I already heard of unprojecting the whole scene but I decided against that because it sounds hacky and isn't accurate. Maybe I could somehow cast a ray in view direction but I do not know how to check the collision with a block in my voxel data. Of course this calculations must be done on the CPU since I need the results to perform game logic operations.  So how could I find out which block is in front of the camera? If it is preferable, how could I cast a ray and check collisions? Solving $\sqrt{x+5} = x - 1$ Known proofs of Wirtinger's Inequality? How does ADR (Automated Dialogue Replacement) work?  There's an entry in Wikipedia that says what it is (essentially actors dubbing their lines in post production, sometimes requiring several takes) but I don't see what's "automatic" about it.  It doesn't seem to work all that well because I can almost always tell when it has been used.