Find and prove formula
Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of natural numbers, order properties of natural numbers, the trichotomy of order for natural numbers and principle of induction.  Proposition 2.2.14 (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m ≥ m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 ≤ m' &lt; m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m ≥ m_0$. Exercise 2.2.5. Prove Proposition 2.2.14. (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0 ≤ m &lt; n$; note that $Q(n)$ is vacuously true when $n &lt; m_0$.)  Thanks! Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) What is the correlation if the standard deviation of one variable is 0? On the bash command-line, ↑ gives me the previous command. On the command-lines in ipython or matlab, when I type a few characters, ↑ gives me the previously entered command starting with those characters. How can I enable exactly this behaviour in bash? I am aware of more advanced ways of searching through the command-line history, but sometimes a simple way is more convenient. Get output parameter value in ADO.NET Can you reduce yourself, crawl into the Tarrasque's airway, and try to collapse its lung by enlarging inside of it? A player tried to use the spell  on his PC while in the mouth of the Tarrasque to shrink himself, then go deeper into its airway and to try to collapse its lung by enlarging himself inside of it. I imagine there's no real rule for this but I felt silly just denying it, because it seemed like a cool idea that you could only probably do with a creature as large as the Tarrasque.  Can you crawl into a gargantuan creature’s mouth or into its airway via reduce/enlarge and attack its brain or internal organs? Fastest way to remove non-numeric characters from a VARCHAR in SQL Server I'm writing an import utility that is using phone numbers as a unique key within the import.  I need to check that the phone number does not already exist in my DB. The problem is that   phone numbers in the DB could have things like dashes and parenthesis and possibly other things. I wrote a function to remove these things, the problem is that it is slow and with thousands of records in my DB and thousands of records to import at once, this process can be unacceptably slow. I've already made the phone number column an index.  I tried using the script from this post:   But that didn't speed it up any.  Is there a faster way to remove non-numeric characters? Something that can perform well when 10,000 to 100,000 records have to be compared.  Whatever is done needs to perform fast.  Update Given what people responded with, I think I'm going to have to clean the fields before I run the import utility.   To answer the question of what I'm writing the import utility in, it is a C# app. I'm comparing BIGINT to BIGINT now, with no need to alter DB data and I'm still taking a performance hit with a very small set of data (about 2000 records).   Could comparing BIGINT to BIGINT be slowing things down?  I've optimized the code side of my app as much as I can (removed regexes, removed unneccessary DB calls). Although I can't isolate SQL as the source of the problem anymore, I still feel like it is. Title does not specify "On Hold" Testing the app, I noticed the title does not specify if a question is "On Hold" ... I haven't checked specifically, but would assume it doesn't specify as "Duplicate" either ... Or am I missing it somewhere? Is it not in the title but located/displayed some other way? If it is, this could be very confusing to users. If it isn't there, it's definitely confusing to users :-)  Using: Version 1.0.1.125 on iPad3 with iOS8 In Bash, two integers can be compared using conditional expression     arg1 OP arg2      OP is one of -eq, -ne, -lt, -le, -gt, or -ge. These   arithmetic binary operators return true if arg1 is equal to, not equal   to, less than, less than or equal to, greater than, or greater than or   equal to arg2, respectively. Arg1 and arg2 may be positive or negative   integers.   or arithmetic expression:     &lt;= &gt;= &lt; &gt; comparison      == != equality and inequality   Why do we have two different ways for comparing two integers? When to use which?  For example, [[ 3 -lt 2 ]] uses conditional expression, and (( 3 &lt; 2 )) uses arithmetic expression. Both returns 0 when the comparison is true  When comparing two integers, can these two methods always be used interchangeably? If yes, why does Bash have two methods rather than one? If I just render the default cube it will look like this:   It is rendered with color depth 16. I can see there are artefacts consisting of visible "line strokes" on some of the cube faces.  The bands seems to not go darker and darker, but alternates between light and dark, or is my eye fooled?  Why?  Edit  Here comes a gradient created by gimp without banding, using the same computer: I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:  $V(1): 1≤1 \text{  true}$  $V(n): n!≤((n+1)/2)^n$  $V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$  and I've got : $(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ $((n+1)^n)n(n+1)≤((n+2)^n)((n/2)+1)$ Can I use the valid Schengen visa in old passport? What exactly is a photon? Consider the question, .  The answers say, "an elementary particle" and not much else. They don't actually answer the question. Moreover, the question is flagged as a duplicate of,  &ndash; the answers there don't tell me what a photon is either. Nor do any of the answers to  mentioned in the comments. When I search on , I can't find anything useful. Questions such as,  look promising, but bear no fruit.  say things like, "the photon is an excitation of the photon field." That tells me  nothing. Nor does the , which says:     The photon is the quantum of the electromagnetic four-potential, and therefore the massless bosonic particle associated with the electromagnetic force, commonly also called the 'particle of light'...   I'd say that's less than helpful because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This same concept is in the  too - but it isn't true. As , "Virtual particles only exist in the mathematics of the model." So, who can tell me what a real photon is, or refer me to some kind of authoritative informative definition that is accepted and trusted by particle physicists? I say all this because I think it's of paramount importance. If we have no clear idea of what a photon actually is, we lack foundation. It's like what :     Photons seem to be one of the foundation ideas of quantum mechanics, so I am concerned that without a clear definition or set of concrete examples, the basis for understanding quantum experiments is a little fuzzy.   I second that, only more so. How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else? It all starts with the photon.   I will give a 400-point bounty to the least-worst answer to the question. One answer will get the bounty, even if I don't like it. And the question is this:    What exactly is a photon? How to both display a command line's output on console and save the output into a text file? What is the effect of having correlated predictors in a multiple regression model? I learned in my linear models class that if two predictors are correlated and both are included in a model, one will be insignificant. For example, assume the size of a house and the number of bedrooms are correlated. When predicting the cost of a house using these two predictors, one of them can be dropped because they are both providing a lot of the same information. Intuitively, this makes sense, but I have a some more technical questions:   How does this effect manifest itself in p-values of the regression coefficients when including only one or including both predictors in the model? How does the variance of the regression coefficients get affected by including both predictors in the model or just having one? How do I know which predictor the model will choose to be less significant? How does including only one or including both predictors change the value/variance of my forecasted cost? How to find orphaned files and images which are not linked from any css stylesheet or from any node? Is there a way to list all unused files which reside inside public file system and inside a theme directory and list them, or maybe even have an option to delete them automatically?  I mean files which are currently not linked from any css stylesheet or from any node. EntityFramework - contains query of composite key When to use self references or composition? How do I check if an object has a specific property in JavaScript? How do I check if an object has a specific property in JavaScript?  Consider:  x = {'key': 1}; if ( x.hasOwnProperty('key') ) {     //Do this }   Is that the best way to do it? Hiding sublayer with toggle button using LayerActions? I have a toggle map (like ) with sublayers. I want all the sublayers to be visible when you open the map, but I would like to hide some sublayers when the toggle buttons are clicked. I've tried adding a hide sublayer code into the LayerActions section of my code, but it's not hiding the sublayers (the button does the same function no matter whether the 'layer.getSubLayer(1).hide()' is there or not). This is the code I'm using:  var LayerActions = {  all: function(){   layer.getSubLayer(1).hide();   sublayers[0].setSQL("SELECT * FROM spreadsheet");   return true;  },  2011: function(){   layer.getSubLayer(1).hide();   sublayers[0].setSQL("SELECT * FROM spreadsheet WHERE year IN ('2011') AND year IS NOT NULL");   return true;  },  2012: function(){   layer.getSubLayer(1).hide();   sublayers[0].setSQL("SELECT * FROM spreadsheet WHERE year IN ('2012', '2012/2013') AND year IS NOT NULL");   return true;  } }   $('.button').click(function() {  $('.button').removeClass('selected');  $(this).addClass('selected');  LayerActions[$(this).attr('id')](); });   Could someone point me to where I'm messing up with my code? Or is it not possible to hide a sublayer using the LayerActions?    I'm now loading my layers differently, and this is my new code:   cartodb.createLayer(map, url1)     .addTo(map, 0)     .done(function (layer) {         console.log('added url1 ',url1);         cartodb.createLayer(map, url2)         .addTo(map, 1)         .done(function (layer) {             console.log('added url2 ',url2);         })         .error(function (){             console.log('problem adding 2nd layer');         });     })     .error(function () {         console.log('There is something wrong with requested data!');     });   So if I wanted to hide the first layer that's loaded (url1) and select only certain points from the second layer (url2), how would I code that in my LayerActions section? I've tried the following, but nothing changes when I click the buttons:  var LayerActions = { all: function(){   layer.getLayer(0).hide;   layer[1].setSQL("SELECT * FROM spreadsheet");   return true; }, 2011: function(){   layer.getLayer(0).hide;   layer[1].setSQL("SELECT * FROM spreadsheet WHERE year IN ('2011') AND year IS NOT NULL");   return true; }, ...