LECTURE NOTES ON MATHEMATICAL INDUCTION PETE L. CLARK Contents 1. Introduction 2. The (Pedagogically) First Induction Proof 3. The (Historically) First(?) Induction Proof 4. Closed Form Identities 5. More on Power Sums 6. Inequalities 7. Extending binary properties to n-ary properties 8. Miscellany 9. The Principle of Strong/Complete Induction 10. Solving Homogeneous Linear Recurrences 11. The Well-Ordering Principle 12. Upward-Downward Induction 13. The Fundamental Theorem of Arithmetic
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Each new generation brings the reemergence of many of the fears of the past‚ requiring the repetition of old explanations to put them to rest. Today there is a renewed concern that technological advancement may displace much of the manufacturing (and other) work force‚ creating widespread unemployment‚ social disruption‚ and human hardship. For example‚ in 1983 the Upjohn Institute for Employment Research forecast the existence of 50‚000 to 100‚000 industrial robots in the United States by 1990‚
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The most amazing place that I ever visited is New York. It is also called the city that never sleeps‚ The capital of the world‚ The big apple or Empire city. This city has many places to visit‚ some of these places are: The famous statue of liberty‚ The Empire State Building‚ Central Park‚ Times Square‚ the popular 5th Avenue‚ and more places. I will talk about those in this essay. Also i will explain why New York is the most amazing city I’ve ever been. I will talk about the places that
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Problem-Solutions Essay Standing Up For Education Education is one of the key ingredients to success. However‚ there is an ongoing problem with education that exists all over America‚ generally‚ amongst minorities in poverty ridden regions. “Education should have no colors or boundaries. Everyone should have an equal opportunity to learn” (Adams 5). The fact of the matter is that there are several things that make this difficult. One major problem is that there is no equality when distributing
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The City So Nice They Named It Twice: New York‚ New York When you associate anything with New York City it is usually the extraordinary buildings that pierce the sky or the congested sidewalks with people desperate to shop in the famous stores in which celebrities dwell. Even with my short visit there I found myself lost within the Big Apple. The voices of the never-ending attractions call out and envelop you in their awe. The streets are filled with an atmosphere that is like a young child on
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works are vastly different‚ with Shakespeare’s play being a more involved study of human nature and psychology. There are‚ however‚ a few deviations from Shakespeare’s source‚ one of which being the motivations of the Iago figure. Cinthio’s Iago was driven to revenge when Desdemona refused to have an affair with him; Iago’s motivations are not nearly so plain in Shakespeare’s version. Shakespeare tired to get the point of jealousy across more then revenge. (Othello) The idea of Othello returning to
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News is presented to the public in numerous mediums‚ including newspapers‚ magazines‚ television programs and the internet. With all of these sources providing information‚ it is obvious that some of the articles or stories will not be as reliable as others. By being news literate‚ capable of analyzing news reports and judging their credibility‚ the news consumer can accurately determine what information should be taken‚ and what should be discarded as opinion and unreliable. Criteria for good
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To: Board of Directors‚ New World Computing Inc. From: Casey Hampton‚ Consultant (Vidhan Jain) Subject: Consultation report Date: Executive Summary: Working on a new‚ innovative and an easy to use product in the form of Pulse in the growing market of medical technology‚ New World Computing has tapped what is a very promising business idea. With limited competition and superior product design‚ the firm has immense potential to mass market its product and to reap the benefits of the
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Courses http://oyc.yale.edu/transcript/469/engl-300 ENGL-300: INTRODUCTION TO THEORY OF LITERATURE Lecture 19 - The New Historicism [March 31‚ 2009] Chapter 1. Origins of New Historicism [00:00:00] Professor Paul Fry: So today we turn to a mode of doing literary criticism which was extraordinarily widespread beginning in the late seventies and into the eighties‚ called the New Historicism. It was definable in ways that I’ll turn to in a minute and‚ as I say‚ prevalent to a remarkable degree
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Mathematical model A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also writtenmodeling). Mathematical models are used not only in the natural sciences (such as physics‚ biology‚ earth science‚ meteorology) and engineering disciplines (e.g. computer science‚artificial intelligence)‚ but also in the social sciences (such as economics‚ psychology‚ sociology and political science); physicists
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