What about the 'geometry' in 'geometric progression'? Every time I have come upon a discussion of the geometric sequence, I have often wondered (in vain) about the qualifier 'geometric' since such sources never explained the origin of the term. Naturally, I have wondered about the related terms 'arithmetic' and 'harmonic' in the name of their respective sequences too; while I have been able to find some plausible explanation of the origin of these latter two terms, I have however wondered on end about the historical origin of the 'geometric' in the term geometric progression, without appreciable success.  You might have come across, or thought of, a plausible connection between exponential sequences and geometry that gave such sequences their collective name. Please share these below.  Thanks plenty.
Why is a geometric progression called so? Just curious about why geometric progression is called so. Is it related to geometry?
What is the highest power of n in $(n^r-1)!$ where $n$, $r$ are positive integers? The answer supplied is $\frac{n^r-nr+r-1}{n-1}$   The highest power of $n$ in $n^r!  = r +$ the highest power of $n$ in $(n^r-1)!$ as $n^r$ contributes $r$ powers.    So,  the highest power of $n$ in $n^r!$ needs to be $\frac{n^r-nr+r-1}{n-1} + r =\sum_{1≤t&lt;r}n^t $.    This is clearly true if $n$ were prime according to .  Again,  if $r=1$, $\frac{n^r-nr+r-1}{n-1}$ becomes $0$.    But according to , the  highest power of composite $n$ in $(n-1)! ≥ 1 $.  So, the answer does not seem to hold for $r=1$ if $n$ is composite. If $n=\prod p_i^{a_i}$, where $p_i$ are distinct primes,  $n^r=\prod p_i^{ra_i}$  So, the highest power $(q_i)$ of $p_i$ in  $n^r!$ is  $$\sum_{1≤t&lt;∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor =\sum_{1≤t≤ra_i}\frac{n^r}{p_i^t} +\sum_{ra_i+1≤t&lt;∞}\left\lfloor\frac{n^r}{p_i^t} \right\rfloor$$  as $p^t|n^r$ for $t≤ra_i$.  So, the highest power of $p_i^{a_i}$ in  $n^r!$ is $\left\lfloor \frac{q_i}{a_i} \right\rfloor=P_i$(say).   So, the highest power of n=$\prod p_i^{a_i}$ in  $n^r!$ is $min(P_i)$  I could not proceed anymore. The following example works as expected:  \documentclass{article} \usepackage[table]{xcolor} \usepackage{makecell, multirow}     \renewcommand\theadfont{\normalsize\bfseries\color{white}}     \renewcommand{\theadgape}{} \usepackage{hhline}  \begin{document} \centering     \setlength\arrayrulewidth{2pt}     \arrayrulecolor{yellow}     \def\clinecolor{\hhline{&gt;{\arrayrulecolor{yellow}}--%                             &gt;{\arrayrulecolor{gray}}-}}     \setlength\extrarowheight{2pt} \begin{tabular}{ c | c | c }%     \rowcolor{gray} \multicolumn{2}{c|}{\thead{AAA -- BBBB}}    &amp;                                   \\     \clinecolor     \rowcolor{gray} \color{white} AAA   &amp;   \color{white} BBB   &amp;   \multirow{-2}{*}{\thead{CCC}}   \\     \hhline{&gt;{\arrayrulecolor{yellow}}-|-|-}     \rowcolor{gray!25} aaa                 &amp;   bbb                 &amp;   ccc                             \\     \hline     \rowcolor{gray!25} ddd             &amp;   eee                 &amp;   fff     \end{tabular} \end{document}     But the following not:  \documentclass[xcolor=table]{beamer} \usepackage{makecell, multirow}     \renewcommand\theadfont{\normalsize\bfseries\color{white}}     \renewcommand{\theadgape}{} \usepackage{hhline}  \begin{document} \begin{frame} \frametitle{test} \centering     \setlength\arrayrulewidth{2pt}     \arrayrulecolor{yellow}     \def\clinecolor{\hhline{&gt;{\arrayrulecolor{yellow}}--%                             &gt;{\arrayrulecolor{gray}}-}}     \setlength\extrarowheight{2pt} \begin{tabular}{ c | c | c }%     \rowcolor{gray} \multicolumn{2}{c|}{\thead{AAA -- BBBB}}    &amp;                                   \\     \clinecolor     \rowcolor{gray} \color{white} AAA   &amp;   \color{white} BBB   &amp;   \multirow{-2}{*}{\thead{CCC}}   \\     \hhline{&gt;{\arrayrulecolor{yellow}}-|-|-}     \rowcolor{gray!25} aaa                 &amp;   bbb                 &amp;   ccc                             \\     \hline     \rowcolor{gray!25} ddd             &amp;   eee                 &amp;   fff                             \\     \hhline{|-|-|-|}     \end{tabular} \end{frame} \end{document}     Why are the in beamer rules so strangely misplaced? Is there some workaround? What's a way to implement a flexible buff/debuff system? Overview:  Lots of games with RPG-like statistics allow for character "buffs", ranging from simple "Deal 25% extra damage" to more complicated things like "Deal 15 damage back to attackers when hit."  The specifics of each type of buff aren't really relevant.  I'm looking for a (presumably object-oriented) way to handle arbitrary buffs.  Details:  In my particular case, I have multiple characters in a turn-based battle environment, so I envisioned buffs being tied to events like "OnTurnStart", "OnReceiveDamage", etc.  Perhaps each buff is a subclass of a main Buff abstract class, where only the relevant events are overloaded.  Then each character could have a vector of buffs currently applied.  Does this solution make sense?  I can certainly see dozens of event types being necessary, it feels like making a new subclass for each buff is overkill, and it doesn't seem to allow for any buff "interactions".  That is, if I wanted to implement a cap on damage boosts so that even if you had 10 different buffs which all give 25% extra damage, you only do 100% extra instead of 250% extra.  And there's more complicated situations that ideally I could control.  I'm sure everyone can come up with examples of how more sophisticated buffs can potentially interact with each other in a way that as a game developer I may not want.  As a relatively inexperienced C++ programmer (I generally have used C in embedded systems), I feel like my solution is simplistic and probably doesn't take full advantage of the object-oriented language.  Thoughts?  Has anyone here designed a fairly robust buff system before?  Edit: Regarding Answer(s):  I selected an answer primarily based on good detail and a solid answer to the question I asked, but reading the responses gave me some more insight.  Perhaps unsurprisingly, the different systems or tweaked systems seem to apply better to certain situations.  What system works best for my game will depend on the types, variance, and number of buffs I intend to be able to apply.  For a game like Diablo 3 (mentioned below), where nearly any bit of equipment can change a buff's strength, the buffs are just character stats system seems like a good idea whenever possible.  For the turn-based situation I'm in, the event-based approach may be more suitable.  In any case, I'm still hoping someone comes along with a fancy "OO" magic bullet which will allow for me to apply a +2 move distance per turn buff, a deal 50% of damage taken back to the attacker buff, and a automatically teleport to a nearby tile when attacked from 3 or more tiles away buff in a single system without turning a +5 strength buff into its own subclass.  I think the closest thing is the answer I marked, but the floor is still open.  Thanks to everyone for the input. I live in an apartment which provides Internet service included in the rent. They use a company called Pavlov for the Internet    wireless seems to be working fine but wired connection is at least %30 faster.  Ethernet, Cat5 outlet is built in the wall, and there is only 1 outlet in each room.  I would like to take this 1 outlet coming out the wall and multiply it into 4 wires, for desktop, Playstation, TV and laptop, without losing any Internet bandwidth.   I have absolutely no idea whether this line is coming from a switch or a router but I have been researching Ethernet splitter, routers, switches, hubs and haven't found a solid answer.   Is there a way I can do this?  This picture cleared up few things:     It seems that an Ethernet switch is to Ethernet as a USB hub is to USB, but with packet inspection.  What is a 10/100Mbps Network Switch, actually, and what is the cap? Wifi problems with rtl8723be in Ubuntu 14.04 Gibbs free energy intuition What is ? As my book explains:     Gibbs energy is the energy of a system available for work.   So, what does it want to tell? Why is it free? Energy means ability to do work. What is so special about this energy? Can anyone simply explain?   I just want a math-free intuition. Theory behind partial least squares regression Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA?  I have looked at many sources online and have not found anything that had the right combination of rigor and accessibility.    I have looked into The Elements of Statistical Learning, which was suggested in a comment on a question asked on Cross Validated, , but I don't think that this reference does the topic justice (it's too brief to do so, and doesn't provide much theory on the subject).  From what I've read, PLS exploits linear combinations of the predictor variables, $z_i=X \varphi_i$ that maximize the covariance $ y^Tz_i $ subject to the constraints $\|\varphi_i\|=1$ and $z_i^Tz_j=0$ if $i \neq j$, where the $\varphi_i$ are chosen iteratively, in the order in which they maximize the covariance.  But even after all I've read, I'm still uncertain whether that is true, and if so, how the method is executed. For every vector space $V$ does there exist a linear functional $f$ ( a linear map from $V$ to $F$ the underlying field ) such that for some $ \vec v \in V$ , $f(\vec v) \ne 0$ ? If it does exist , can we prove the existence without the "axiom of choice " ? Is the existence equivalent to axiom of choice ? Access a custom .NET DLL in VBScript I wrote a DLL in .NET and I want to access it in VBScript. I don't want to add it to the assembly directory.   Is there a way to point too the DLL and create an instance of it? What should main() return in C and C++? What is the correct (most efficient) way to define the main() function in C and C++ — int main() or void main() — and why? And how about the arguments? If int main() then return 1 or return 0?  There are numerous duplicates of this question, including:       — For C++, with a very good answer indeed.     Related: How to find and replace a string by increasing its numerical part? Remove appx package for all users Fresh Windows 10 install. Somehow the MSN Weather app became corrupted. (Unable to launch) I removed the app with the following Powershell command:     Remove-AppxPackage -Package Microsoft.BingWeather_4.7.118.0_x86__8wekyb3d8bbwe   However, when running     Get-Appxpackage –Allusers   the package is still present. When I attempt to reinstall the app via the Windows Store, the download is skipped and the same corrupted package is installed.  I've tried restarting (many times) and using WSReset to no avail.  What can I do? Thank you! Preserving signatures of decorated functions Suppose I have written a decorator that does something very generic. For example, it might convert all arguments to a specific type, perform logging, implement memoization, etc.  Here is an example:  def args_as_ints(f):     def g(*args, **kwargs):         args = [int(x) for x in args]         kwargs = dict((k, int(v)) for k, v in kwargs.items())         return f(*args, **kwargs)     return g  @args_as_ints def funny_function(x, y, z=3):     """Computes x*y + 2*z"""     return x*y + 2*z  &gt;&gt;&gt; funny_function("3", 4.0, z="5") 22   Everything well so far. There is one problem, however. The decorated function does not retain the documentation of the original function:  &gt;&gt;&gt; help(funny_function) Help on function g in module __main__:  g(*args, **kwargs)   Fortunately, there is a workaround:  def args_as_ints(f):     def g(*args, **kwargs):         args = [int(x) for x in args]         kwargs = dict((k, int(v)) for k, v in kwargs.items())         return f(*args, **kwargs)     g.__name__ = f.__name__     g.__doc__ = f.__doc__     return g  @args_as_ints def funny_function(x, y, z=3):     """Computes x*y + 2*z"""     return x*y + 2*z   This time, the function name and documentation are correct:  &gt;&gt;&gt; help(funny_function) Help on function funny_function in module __main__:  funny_function(*args, **kwargs)     Computes x*y + 2*z   But there is still a problem: the function signature is wrong. The information "*args, **kwargs" is next to useless.  What to do? I can think of two simple but flawed workarounds:  1 -- Include the correct signature in the docstring:  def funny_function(x, y, z=3):     """funny_function(x, y, z=3) -- computes x*y + 2*z"""     return x*y + 2*z   This is bad because of the duplication. The signature will still not be shown properly in automatically generated documentation. It's easy to update the function and forget about changing the docstring, or to make a typo. [And yes, I'm aware of the fact that the docstring already duplicates the function body. Please ignore this; funny_function is just a random example.]  2 -- Not use a decorator, or use a special-purpose decorator for every specific signature:  def funny_functions_decorator(f):     def g(x, y, z=3):         return f(int(x), int(y), z=int(z))     g.__name__ = f.__name__     g.__doc__ = f.__doc__     return g   This works fine for a set of functions that have identical signature, but it's useless in general. As I said in the beginning, I want to be able to use decorators entirely generically.  I'm looking for a solution that is fully general, and automatic.  So the question is: is there a way to edit the decorated function signature after it has been created?  Otherwise, can I write a decorator that extracts the function signature and uses that information instead of "*kwargs, **kwargs" when constructing the decorated function? How do I extract that information? How should I construct the decorated function -- with exec?  Any other approaches? I have a "term reference" field, in a form that uses an auto-complete widget. I want to limit user input to terms in the vocabulary only, and output a message when a user tries to enter a non existing term.  I found the  module. However this does not work with forms that contain a field collection or a field group.  There is also a contributed patch . But I was wondering whether there was a simpler/cleaner solution like a module or maybe I missed an already existing way of doing it with latest drupal and taxonomy versions. How to install HPLIP on my Ubuntu to support my HP printer and/or scanner? I'm buying a bus ticket to get to Boston 23 hours before my flight to ensure I reach my departure airport on time.  If I didn't arrange bus transport well in advance, what legal remedies would I have if a delayed bus caused me to miss my flight? Can helium disappear from Earth? Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ The binomial expansion:  $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$  The General Leibniz rule (used as a generalization of the product rule for derivatives):  $$(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}$$  Both formulas can be obtained simply by induction; Newton's binomial also has a combinatorial proof . It's striking how these formulas are similar; is there a possible connection between them? I was thinking that maybe the General Leibniz rule could be obtained using a combinatorial argument as well (hence the binomial coefficients)...