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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1. Start off by weighing two paracetamol tablets using some accurate scales. Record the mass down (1.15g). This had to be as accurate and precise as we could get it so we had the correct weight to start the experiment. Without this then the results we found at the end would not be accurate. A problem that could be faced with this is that the scales may not have been fully set to zero. Also it takes time for it to fully reach its weight on the reading so you may move them too quick before the right
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NH3 H2O - NH4OH can be a very useful to the society because by keeping an eye out for this synthesis reaction‚ chemical technicians are able to identify ammonium hydroxide in water accessed by people and prevent people from being in contact with toxic water. Ammonia takes the form of gas at room temperature. It is a colourless gas with an irritating odour. Ammonias melting point is at -77 degrees Celsius and its boiling point is at -33.35 degrees Celsius. This covalent compound is corrosive to
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My hypothesis is that if I turn the water to vapor then I will be able to determine the ratio and the amount of moles of the salt and water. This is because by using evaporation I will be able to learn the mass of the salt and of the water needed for n=m/MM which states the amount of moles can be found by dividing mass of the moles by the molar mass. Experiment 1 First I took a crucible‚ a balance‚ and a Bunsen burner and placed them onto the workbench. Then I measured the mass of the empty crucible
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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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