Proving $1 + 2 + 2^2+2^3 + 2^4 + \dots+ 2^{n-1}= 2^n -1$ using mathematical induction
Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction?
If N and every subgroup of N is normal in G then G/N is abelian . Let $N$ be a normal subgroup of a group $G$ such that every subgroup of $N$ is normal in $G$ and $C_G(N)\subseteq N $. Prove that $G/N$ is abelian.   Here, as usual, $C_G\left(N\right)$ means the centralizer of $N$ in $G$ (i.e., those elements of $G$ that commute with everything in $N$).  I think we need to use that every subgroup of $N$ is normal in $G$ but i can't use .Please help me with Hints. What is the best technique for long-exposure photography in good light? How do I select an object in the 3D viewport? If a company is bought by another company, does my stock rollover to the new company when it is combined with the new parent company? How can I disable the "no backup in x days" from TimeMachine? (Updated answer choice for High Sierra. See the old answer if you are still on Yosemite)  I'm running OS X Yosemite, and I want to disable the notifications from Time Machine. I rarely do a hard-drive backup because all of my essential files are in the cloud. I don't need to run timemachine as often as it suggests/warns/annoys me about.  How do I disable the notifications? Time Machine is not in the Notifications preferences, and I don't want to stop using it entirely. CUDA Driver API vs. CUDA runtime I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof:     Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X, \mathcal{M})$, then the functions:      $$g_{1}(x)=\sup_{j} f_{j}(x)$$ $$g_{2}(x)=\inf_{j} f_{j}(x)$$ are   measurable.       Proof:      We have       $$g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}  f_{j}^{-1}((a,\infty])$$ $$g_{2}^{-1}([-\infty,a))=  \bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$$      Where the result follows from the fact that $g_{1}^{-1}((a,\infty])\in  \mathcal{M}$ for all $a \in \mathbb{R} \iff g$ is measurable, and   $g_{2}^{-1}([-\infty,a))\in \mathcal{M}$ for all $a \in \mathbb{R}  \iff g$ is measurable.   I understand the last part of the proof, but how do we know $g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}f_{j}^{-1}((a,\infty])$ and $g_{2}^{-1}([-\infty,a))=\bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$? I would appreciate any help with this. Protect .NET code from reverse engineering? Obfuscation is one way, but it can't protect from breaking the piracy protection security of the application. How do I make sure that the application is not tampered with, and how do I make sure that the registration mechanism can't be reverse engineered?  Also it is possible to convert a C# application to native code, and  is too costly.  C# provides lot of features, and is the ideal language for my code, so writing the whole codebase again in C++ is out of the question.  Secure certificates can be easily removed from the signed assemblies in .NET. JSON Parsing in Android Let $p, q \in \mathbb R$ be positive reals for which $p + q = 1$. How to prove that for any two natural numbers $n, m$ the following inequality is true?  $(1 - p^n)^m + (1 - q^m)^n \ge 1$  I don't have a big knowledge about solving inequalities, so I tried to use cauchy-schwarz inequality, binomial theorem and some other baisc techniques, but it lead me nowhere. I've been thinking about it for a long time and now I'm completly stuck. Scala final vs val for concurrency visibility Subsample of a random sample: random sample? Show that the conditional statement is a tautology without using a truth table I have been attempting to use identities to get to the answer but I am unable to get anywhere. Here is the equation I am trying to prove tautological without using truth tables:  $[(p\rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r )$  And here is my attempt at using identities to work my way up. (a link to a page with a table of logical equivalence identities: )  $\lnot [(p \rightarrow q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $\lnot [(\lnot p \lor q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $[\lnot (\lnot p \lor q) \land \lnot (q \rightarrow r)] \lor (p \rightarrow r)$  $[(p\land \lnot q) \land (q \land \lnot r)] \lor (p \rightarrow r)$  I don't know where to go from there. I'm starting out in Visual C++ and I'd like to know how to keep the console window.  For instance this would be a typical "hello world" application:   int _tmain(int argc, _TCHAR* argv[]) {     cout &lt;&lt; "Hello World";     return 0; }   What's the line I'm missing? Why is python ordering my dictionary like so? Why did my electric range wiring burn out at the terminal block? Turned on our 2 year old stove (Whirlpool Gold) and heard fizzing noises, then smelled electrical burning. Went downstairs and shut off the breaker (40 amp). That was the only way to get it to stop "fizzing". (6 gauge wire by the way). The range is 240 volt electric 3 wire hooked up to a 40 amp breaker (6 gauge wire). When I took the back panel off the stove I saw exactly where the problem occurred. The black wire was burned, the terminal block was melted, and the inside of the back metal panel that I removed was pitch black in the spot that was facing that area before removal. I removed the stove top and inspected all wires and other electronic components. No problems at all.   So my questions are:    Why did the black wire only, at the terminator block, start burning?  Why did the breaker not pop off?  Is it safe to just replace the terminator block or is this an indicator of a much larger problem?  Is 6 gauge the correct gauge for a 40 amp breaker? Why is RAM not put on the CPU chip? Modern CPUs are very fast compared to all things external, including memory (RAM).   It is understandable, since CPU clock frequency has reached a point where it takes several clock ticks for an electric signal simply to run from from the CPU through the bus to RAM chips and back.   It also complicates life on many levels: multi-level cache hierarchies are built to deliver data closer to the CPU, which in turn require complex synchronization logic in the chip. Programs must be written in a cache-friendly way to avoid wait cycles while data is fetched.   Many of these problems could be avoided if a significant amount of RAM was located directly on the CPU chip. It doesn't have to an exclusive arrangement: maybe put 1-4 GB on the chip, depending on its class and allow additional memory installed separately.   I'm sure there are good reasons Intel, AMD and the like are not doing this. What are these reasons? Is it that there's no room to spare on the chip? Does anyone know how to make the character picker appear more quickly?   I use it enough now in my German language studies that it has become second nature for the critical marks and much faster than using the alt+keypress shortcuts.   Only problem is that it's so second nature that sometimes it doesn't pop up quickly enough so I end up typing the character and the number shortcut sometimes (IE u2 instead of ü). Anyone know how to make it appear with less delay? Ideally, I'd love if I could just hold the character and press the number but it appears that OSX insists on waiting for the menu to pop up. I want to get random decimal digits out of a 6-faced die.  Let's say the die is fair, with an equal probability for each outcome. (But feel free to share any consequences of that not being true, if you feel like it.)  I will represent the outcome of a die roll as a base-6 number:   In base-6 there are only 6 possible digits: $\{0,1,2,3,4,5\}$. The die has faces numbered 1 to 6. Face $f$ corresponds to digit $f-1$.   Thus, I can represent as many die rolls as I like with a base-6 number:   If the die is rolled once:   Let's say the die stopped on face $5$. The final base-6 number will be $4$.  If the die is rolled twice:   Let's say the die stopped on face $5$. And then on face $3$. The final base-6 number will be $42$.  If I roll the dice 10 times:   Let's say the die stopped sequentially on faces $5$, $3$, $4$, $6$, $2$, $3$, $3$, $3$, $2$, and $4$. The final base-6 number will be $4235122213$.    Since I want decimal digits, I need to convert to base-10.   Base-6: $4235122213$ Base-10: $44753985$   For any given number of die rolls, how many different combinations exist? If there are $r$ die rolls, the total of possible combinations is $6^r$.   $r$ rolls — $6^r$ different combinations. $0$ rolls — only $1$ combination. $1$ roll — $6$ different combinations. $2$ rolls — $36$ different combinations. $3$ rolls — $216$ different combinations. $4$ rolls — $1296$ different combinations. $5$ rolls — $7776$ different combinations. $6$ rolls — $46656$ different combinations. $7$ rolls — $279936$ different combinations. $8$ rolls — $1679616$ different combinations. $9$ rolls — $10077696$ different combinations.   $n_r$ ($\in \mathbb{N_0}$) is the number obtained after $r$ die rolls through the process described previously: "If the die is rolled once, twice, ..."  It is impossible for $n_r$ to reach the number of possible different combinations ($6^r$). It will be always short by least $1$:  $$ 0 \leqslant n_r &lt; 6^r $$   $r$ rolls — $n_r$ never surpasses $6^r-1$. $0$ rolls — $n_r$ never surpasses $0$. $1$ roll — $n_r$ never surpasses $5$. $2$ rolls — $n_r$ never surpasses $35$. $3$ rolls — $n_r$ never surpasses $215$. $4$ rolls — $n_r$ never surpasses $1295$. $5$ rolls — $n_r$ never surpasses $7775$. $6$ rolls — $n_r$ never surpasses $46655$. $7$ rolls — $n_r$ never surpasses $279935$. $8$ rolls — $n_r$ never surpasses $1679615$. $9$ rolls — $n_r$ never surpasses $10077695$.   I can never trust the leftmost nor the rightmost digit (fringe digits), since the possible values are potentially restricted.   For $0$ rolls, the outcome is always $0$. For $1$ roll, the fringe digit never surpasses $5$. For $2$ rolls, the leftmost digit never surpasses $3$, and the rightmost, $5$. For $3$ rolls, the leftmost digit never surpasses $2$, and the rightmost, $5$. For $4$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$. For $5$ rolls, the leftmost digit never surpasses $7$, and the rightmost, $5$. For $6$ rolls, the leftmost digit never surpasses $4$, and the rightmost, $5$. For $7$ rolls, the leftmost digit never surpasses $2$, and the rightmost, $5$. For $8$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$. For $9$ rolls, the leftmost digit never surpasses $1$, and the rightmost, $5$.   The other digits seem perfectly random to me though. Am I wrong? Are there 'stack' sites for Mac or Windows?