How do I prove $\sum_{k=0}^{n}\binom{n}{k}\cdot 2^k = 3^n$?
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
How to run adb and fastboot commands from Termux? How do I know when to use "the" versus "a" versus "∅" as an article on a noun? With proper nouns, we don't use the except for river names, newspaper names, etc.  I want to know why we use the with White House. I mean, under which rule can we categorize it? What could other similar examples for that rule be? How to evaluate this infinite sum?  $$\sum_{n=1}^{\infty}\frac{1}{2^n-1}$$ Can we have threaded comments? The conversation often continues in the Comments and important information is held in the comments. I'd like to have threaded comments there. Proof a set cannot contain its own powerset - Russell's Paradox Using naive set theory and not using the cardinal number of a power set, how can you prove a set cannot contain its own power set? Is this connected to Russell's Paradox? Is spell level a character concept or a player concept? Is there anything in RAW that supports spell level as a character concept, or alternatively that supports it as solely a player concept?  On the one hand, I see nothing that explicitly states that a character knows that there are spell levels. On the other hand, I don't see how a spellcaster could manage spell slots without knowing about spell levels. What is tail recursion? Whilst starting to learn lisp, I've come across the term tail-recursive. What does it mean exactly? What does $2^x$ really mean when $x$ is not an integer? Clear item with custom name? How can I reset my menus? I accidentally opened too many menus (or windows) and I want to reset them to default.  It is really annoying, and effecting my animation.  Thanks Missing settings in GNOME control center installed in Ubuntu MATE I installed GNOME Control Center following  to access Google Drive in Ubuntu MATE 16.04 LTS (MATE desktop environment 1.12.1)   The problem is GNOME Control Center is only showing language support, printers, firewall setup and backups... not online accounts or other settings.  This is what I get:  ~$ gnome-control-center gnome-online-accounts   ** (gnome-control-center:5744): WARNING **: Ignoring launcher landscape-client-settings (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Invalid categories System;Settings; for panel software-properties-gtk.desktop  ** (gnome-control-center:5744): WARNING **: Ignoring launcher ubuntuone-installer (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Could not find settings panel "gnome-online-accounts"   Any ideas?  Thanks! I have functionality on a cloud page that depends on the cloud page url. I know how to Requestparameter(), but how do I get the entire url using ampscript or SSJS? Keytool Signing Problem: Keystore was tampered with, or password was incorrect If you cast Blindness/Deafness as a 3rd level spell, can you target one creature with both effects? Inequality involving the integral of vector valued function Suppose $f :[a,b]\to \mathbb{R}^n$ is given by $f(t)=\langle f_1(t), \ldots, f_n(t)\rangle$, where each of the $n$ component functions is integrable over $[a,b]$. I think the following inequality   $$\left|\int _a^b f(t) \, dt\right|\leq\int_a^b\left|f(t)\right| \, dt$$  still hold but I don't know how to prove it. Any idea?  Thanks Expected number of games played in a game with 'win by two' rule. Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two”” rule: the first player to win two games more than his opponent wins the match. What is the expected number of games played? My attempt: I use Markov chains. Let $S$ denote that starting state, $W$ and $L$ denote the states where Calvin and Hobbes win respectively. Then, I have $$\mu_S = 1+ p\mu_W+q\mu_L $$ $$\mu_W = 1+ q\mu_S $$ $$\mu_L = 1+ p\mu_S $$ where $\mu_j$ denote the expected number of games from state $j$. Solving the system, I get the solution $\mu_S = \frac{2}{1-2pq}$ which I think is correct. I however would like to know how to approach it with geometric distribution instead of Markov chains. I read one solution where the number of games is given as $1+\text{Geom}(p^2+q^2)$ but I do not understand it. My understanding is that for the game to stop, the required probability is $p^2 + q^2$ since this is the probability that either Calvin or Hobbes win two games in a row. How to use this information in a Geometric random variable? Thanks. In the following sentence:     The city is populated by many people.   "many" can be replaced by something like "a great number of" (let us leave the perceived differences between the meanings of these two modifiers out of discussion):     The city is populated by a great number of people.   I understand that this replacement does not change the object "populating the city" from the original "people" to a "number", but rather shows the ability of expressions in the form "[numeral] of" to provide information about the quantity of nouns they modify.  Is this ability due to some special qualities of nouns such as "number", "quantity", "fraction", &amp;c., or is it just one usage of the preposition "of"? Using Moq to determine if a method is called It is my understanding that I can test that a method call will occur if I call a higher level method, i.e.:  public abstract class SomeClass() {         public void SomeMehod()     {         SomeOtherMethod();     }      internal abstract void SomeOtherMethod(); }   I want to test that if I call SomeMethod() then I expect that SomeOtherMethod() will be called.   Am I right in thinking this sort of test is available in a mocking framework? Do I need to activate the firewall? I only use Ubuntu for a home desktop use? I am a novice at Ubuntu (12.04).  I have it on my netbook which I use mainly for surfing the net, learning python and remote desktop into my work computer.    Do I need to enable a Firewall type software?  If so, what would you suggest, and what are sensible default suggestions? Plain integer partitions of $n$ using $r$ parts Division of number $n$ on parts $a_1,...,a_r$ where $a_1 \le ... \le a_r$ we call a plain if $a_1 = 1$ and $a_i - a_{i-1} \le 1$ for $2 \le i \le r$. Find enumerator (generating function) for plain divisions.  my try  The hint was to use bijection between plain divisions and some commonly know enumerator. I tried to use enumerator of divisions on different parts: $$ (1+x)(1+x^2)...(1+x^r)$$ where number of plain divisions is  $$[x^n](1+x)(1+x^2)...(1+x^r) $$ let function  $$f(n,r) = [x^n](1+x)(1+x^2)...(1+x^r) $$  For some first divisions it works. For example: $$f(4,3) = 1 $$ $$f(6,3) = 1 $$ $$f(11,5) = 2$$ But when I tried to find bijection, I failed. I found that this function isn't correct because $f(15,6) = 4$ but should be equal to $3$ because: $$15 = 1,1, 2, 3, 4, 4 \\ 15 = 1, 2, 2, 3, 3, 4\\ 15 = 1, 2, 3, 3, 3, 3 $$. There I stucked.