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Prove that  $ n &lt; 2^{n}  $ for all natural numbers $n$.  I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k&lt;2^{k}$. Considering $k+1 &lt;2^{k}+1$, but where do I go from here?  Any other methods maybe? How can I get my external IP address in a shell script? I have had several business loans and understand that this is not by any means typical.  I noticed that for a year my payments were not decreasing the principal on the loan. When I asked the bank why, I found out that at some point my payments were getting applied to principal only. Now I am paying interest off that was not paid for 6 months or so.  My banker states that he can change all the interest payments to principal payments. I am confused to why this would be. If I make all my principal payments first then would the bank lose a substantial amount of interest?  This has happened on 2 of the 3 loans I have looked at so far.  If they are willing to change all the payments to principal only, would this be beneficial to paying less interest ultimately? I assume there would be an interest account accruing somewhere that would need to be paid off at the end of the loan.  I understand that typically you have a loan balance that would include interest and principal. My balance owed did not change when I continued to make same payment? It gets applied to interest. The balance on the statement must not be a true balance. There must be interest owed above the balance showed but why would they do this to start with? They applied full payments to principle only first in order for interest to get to be more than several payments What would be the the mathematical equivalent of this excel formula? =PMT() I need to create a JavaScript form that does the same calculation as this =PMT() function.   mortgage = 220000 rate= 4.75% term = 30   The example formula I have is =PMT(4.75%/12,30*12,220000*-1)  What would be the equation for this function? I tried to look up this function but it doesn't explain it very well. What is in the space between a nucleus of an atom and its electrons? How to make two people play at the same time? My brother owns Skyrim on steam. We use Steam Family Share, but we were wondering how both of us could play it at the same time? How to find windows 10 activation Key Where can it be find?  At activation panel i can only find option to change the activation key. The current key is not presented on this screen. Combine and Minify Multiple CSS / JS Files Placing math environments inside math environments I have a macro which appears throughout my document in various places:  \newcommand{\placefraction}[2]{     $\frac{#1}{#2}$ }   Sometimes it is placed with regular text. Sometimes it is placed within other math modes. However, the document won't compile if it is placed inside another math mode.  None of my math ever appears in displayed formulas, it is always part of the paragraph.   How can I have a macro which sometimes appears inside math mode, sometimes outside, but itself contains math mode? $\newcommand{\P}{\mathbb{P}}$Is it valid to interpret $\P(B|A)$ as "infinite probability" when $\P(A) = 0$? How can I import Opera bookmarks into Chrome? Our DM keeps interrupting our long rests In the latest campaign, we're still around the third level, but every single night we've spend in the wild so far, we were interrupted by hard combat encounters (as in DMG-worded "hard"), forcing us to use our resources (like HP, spell slots, Ki, once-per-day stuff) and not restoring any of those lost the previous days. I feel like the , as it lasts four to six rounds, so we usually say that we'd like to resume our rest but the answer is always "you've been interrupted, your rest is over, you can't restart any until dusk". So the only real long rests we could have always were in town.  We've always used watch-turns, and being interrupted in the first night of the campaign, we now always spend one hour to find a safe place to rest and we make that intent very clear to the DM. But yet, kobolds, goblins, ghouls and gnolls always find us in the middle of the night, but never when we travel for ten hours during the day (which is comparable to "fast travel" in his own word). Once on the second day in the wild, we were faced to a deadly encounter (as in DMG-worded "deadly") which we survived only by fleeing. And we're kind of forced to get back to the deadly encounter, which is more than one day away from the nearest town.  We're five players and I'm the only one with PnP RPG experience... from ten years ago. I don't remember being this resourceless. We asked if it was a survival campaign and the answer was that it was only a normal introduction campaign found on the Internet that he spiced up a bit. We've also asked if we did anything wrong and the answer was that we followed the normal story.  The DM is mastering his second campaign (the first one is still ongoing, but with veteran players). The DM also plays a lot of survival campaigns as a player.  Is this situation normal? If yes, why? If not, what should I say to the DM?  Note that I'm fully aware of the "long rest" rule as written on page 186 of the PHB. This is why I encouraged my team to say that we resume our rest. (thanks Bloodcinder) table of contents no word separation Is it possible to have in the toc no word separation but have a word separation in the chaptername:  For example:   Contents  1. max Mustermann   Chapter  Max Muster- mann Trying to search for multiple file extensions in command line I am trying to find two extensions in command line but when i do it will only return one of the extensions. can you tell me what I am missing?   Here is my command;  [root]# find /etc -name "*.txt" -o -name "*.log" -type f -mtime +7 -exec echo {} \; Vortex in liquid collects particles in center At xmas, I had a cup of tea with some debris at the bottom from the leaves.  With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin.  At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.  What's the physical explanation of the accumulation of particles in the middle?  My guess is that it's something to do with a larger radius costing the particles more work through friction... Anyone know a simple method to swap the background color of a webpage using JavaScript? How many named characters died in Avengers: Infinity War? How many named characters have died       and/or disintegrated      by the end of Avengers: Infinity War. It is difficult to speculate who died on         Xandar and the Asgardian ship.       So restrict it to on-screen deaths only. Filter subscription email is not being delivered (June 2019 - Feb 2020) I have created a custom filter in  and specified that I would like to get email updates:  However, I never got a confirmation mail nor any notifications based on my filter. I tried the &quot;resent confirmation mail&quot; button on the right hand sidebar:  But again nothing happens. The mail address should be fine  I used it for all kinds of notifications from Stack Exchange. For example the rep recalculation email arrived just fine on this address.  I retyped it several times to make sure there really isn't any typo  I double checked that it did not end up in the spam filter  I don't think it is a problem with my particular email provider, users with other mail providers seem to have the same problem, see     This is possibly related to , but this claims to be &quot;status complete&quot;.   EDIT: Although the &quot;status-completed&quot; tag was added, the problem still persists  EDIT 2: As of Feb 2020, it finally works How do I call one constructor from another in Java? How can I evaluate $\int_{-\infty}^{\infty} e^{-x^2} dx$ without using polar coordinates