Exercise 4.1 2. Establish each of the following for all n ≥ 1 by the Principle of Mathematical Induction. Solution a) S(n): ==‚ S(1): = = =1‚ So S(1) is true. Assume S(k): = Consider S(k+1) = = +=-1+= -1. Hence‚ it follows that S(k)⇒S(k + 1) is true for all n ∈ Z+ by the Principle of Mathematical Induction. b) S( n) for n=1‚ = 2 = 2+(1-1). So S(1) is true. Inductive Step: assume S(k)is true‚ for some (particular) k ∈ Z+—that is‚ assume that =2+(k-1). For n=k+1‚ = + (k+1)
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Working with algebra you must understand why the properties of real numbers are so important. I will demonstrate my solutions to three problems and I will include my mathematical work. Also I will use five vocabulary words that help me find solutions to the problems. Properties of real numbers are useful for simplifying algebraic expression because a lot of thing we do in life are equations. We use a lot of mathematical terms in the real world. Lastly I will show every step I took to simplify and
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Name of the school Grade: ---- Chapter Name – Work Sheet number ------------------------------------------------- Date: -------- Time: Estimated Time to Complete ------------------------------------------------- Name of Student: Date of Submission: Maths assignment XII RELATIONS AND FUNCTIONS 1. If fx= x and x= x ‚ then evaluate fog52-gof-72. 2. If fx= 3-x313 and x= logex ‚ find fof(x). 3. Let * be a binary
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Formal Analysis of Art ART/101 July 17‚ 2011 Formal Analysis of Art There are a lot of differences between expressive and analytical lines in art. Analytical lines are mathematically rigorous‚ precise‚ controlled‚ logical‚ and rational. Sol LeWitt’s Wall Drawing No. 681 is one of the examples of analytical line. (Sayre‚ 2010). Expressive lines are less logical‚ free‚ loose‚ and somewhat out of control. Vincent Van Gogh’s painting “The Starry Night” is an example of art using expressive lines
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The following example shows all three artistic devices combined in the form of an early American political caricature published in New York in 1860 (Graph.01). Following the basic form of artistic interpretation as developed by Erwin Panofsky‚ the first step will be a description of what can be seen inside the image‚ followed by the examination of their iconographical meaning‚ and finally‚ the iconological interpretation will be brought to the light (Panofsky ???). Once these steps are done‚ the
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For Europe‚ the nineteenth century was an age of radical change. As science and technology challenged old views‚ political revolution challenged the old dominations of church and monarchy with the upheaval of the French Revolution of 1789‚ a sequence of revolution‚ counter-revolutions‚ and civil wars in Europe and America continued the assault upon established power. In a world that was experiencing a population explosion of extreme magnitude‚ revolution followed revolution‚ including the industrial
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Written Assignment 1 Section 1.1: 8b). I bought a lottery ticket this week OR I won the million dollar jackpot on Friday. e). I bought a lottery ticket this week IF AND ONLY IF I won the million dollar jackpot on Friday. f).If I did not buy a lottery ticket this week‚then I did not win the million dollar lottery on Friday. 12f).You have the flu AND you miss the final examination‚OR if you do not miss the final examination AND ytou pass the course. 18a). The conditional statement:if
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Running Head : SIMPLIFYING ALGEBRAIC EXPRESSIONS 2 Simplifying Expressions Be sure to have a centered title on page 1 of your papers . [ The introductory paragraph must be written by each individual student a nd the content will vary depending on what the student decides to focus on in the general information of the topic. YOUR INTRODUCTION SHOULD CONNECT MATH CONCEPTS AND REAL - WORLD APPLICATIONS. DO NOT INCLUDE THE DIRECTIONS IN THE INTRO! The following paragr aph is not an
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MA2030 589 UNIVERSITY OF MORA TUW A Faculty of Engineering Department of Mathematics B. Sc. Engineering Level 2 - Semester 2 Examination: MA 2030 LINEAR ALGEBRA Time Allowed: 2 hours 2010 September 2010 ADDITIONAL MATERIAL: None INSTRUCTIONS TO CANDIDATES: This paper contains 6 questions and 5 pages. Answer FIVE questions and NO MORE. This is a closed book examination. Only the calculators approved and labeled by the Faculty of Engineering are permitted. This examination
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Eigenvectors and eigenvalues of a matrix The eigenvectors of a square matrix are the non-zero vectors which‚ after being multiplied by the matrix‚ remain proportional to the original vector‚ i.e. any vector that satisfies the equation: where is the matrix in question‚ is the eigenvector and is the associated eigenvalue. As will become clear later on‚ eigenvectors are not unique in the sense that any eigenvector can be multiplied by a constant to form another eigenvector. For each
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