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Obviously, the theorem is true for N = 1 because both the left-hand and right-hand sides evaluate to 1 Further checking shows that it is true for 2 5 N I 10 However, the fact that the theorem holds for all N that are easy to check by hand does not imply that i t holds for all N Consider, for instance, numbers of the form 2 2 5 1 The first five numbers (corresponding to 0 I k I 4 are 3, 5 , 17, 257, and 65,537 These numbers are all prime Indeed, at one time mathematicians conjectured that all numbers of this form are prime That is not the case We can easily check by computer that 22' + 1 = 641 x 6,700,417 In fact, no other prime of the form 22\ 1 is known A proof by induction is carried out in two steps First, as we have just done, we show that the theorem is true for the smallest cases We then show that if the theorem is true for the first few cases, it can be extended to include the next case For instance, we show that a theorem that is true for all 1 I N I k must be true for 1 I N I k + 1 Once we have shown how to extend the range of true cases, we have shown that it is true for all cases The reason is that we can extend the range of true cases indefinitely We use this technique to prove heorem 81.

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Why does this constitute a proof First, the theorem is true for N = 1, which is called the basis We can view it as being the basis for our belief that " the theorem is true in general In a proof by induction the basis is the easy case that can be shown by hand Once we have established the basis, we use inductive hypothesis to assume that the theorem is true for some arbitrary k and that, under this assumption, if the theorem is true for k, then it is true for k + 1 In our case, we know that the theorem is true for the basis N = 1, so we know that it also is true for N = 2 Because it is true for N = 2, it must be true for N = 3 And as it is true for N = 3, it must be true for N = 4 Extending this logic, we know that the theorem is true for every positive integer beginning with N = 1 Let us apply proof by induction to a second problem, which is not quite as simple as the irst First, we examine the sequence of numbers 12, 2 2 - 1 2 , 3 2 - 2 2 + 12, 4 2 - 3 2 + 2 2 - 1 2 , 5 2 - 4 2 + 3 2 - 2 2 + 12, and s o o n Each member represents the sum of the first N squares, with alternating signs The sequence evaluates to 1, 3,6, 10, and 15 ~ h u sin general, the sum seems , to be equal to the sum of the first N integers, which, as we know from Theorem 81, would be N ( N + 1 )/2 Theorem 82 proves this result.

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0; }. Bar Code Maker In .NET Using Barcode printer for ASP . Make Code-39 In .NET Framework Using Barcode generator for .Now we need to add a reference to the Dojo Toolkit to our page Usually we would do this by downloading the source from the Dojo web site and putting it on our own site then linking to it But one of the goals of this tutorial is to e as simple as possible, so we re going to take advantage of a cool technique for referencing the Dojo source files on the Internet without requiring us to have the source on our own web server AOL provides a facility it calls the Content Delivery Network (CDN), which is a worldwide geographic edge caching mechanism for the InternetThis allows super fast delivery of files to web users from AOL servers that are geographically close to them The files are also compressed, which further improves the download speeds AOL has generously made this facility available to developers and end users For more information on the AOL CDN and Dojo, visit .Related: Java QR Code Generating Data, QR Code Generating Excel Size, QR Code Generation ASP.NET

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Stacks and Compilers. Encode Uniform Symbology Specification Codabar . NET Control to generate, create barcode image in NET applications.In this chapter we examined two uses of stacks in programming language and compiler design We demonstrated that, even though the stack is a simple structure, it is very powerful Stacks can be used to decide whether a sequence of symbols is well balanced The resulting algorithm requires linear time and, equally important, consists of a single sequential scan of the input Operator precedence parsing is a technique that can e used to parse infix expressions It, too, requires linear time and a single sequential scan Two stacks are used in the operator precedence parsing algorithm Although the stacks store different types of objects, the generic mechanism (templates) allows the use of a single stack implementation for both types of objects.Related: EAN 128 Generator ASP.NET , Print ISBN ASP.NET , Create Code 39 .NET WinForms

How to install BarCode SDK Control for SQL Server 2005 Reporting Service (SSRS) and add BarCode Custom Report Control icon to Visual Studio 2005 toolbox. .Related: 

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