Problem_Set_04

Problem_Set_04 - \documentclass[12pt]cfw_amsart...

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\documentclass[12pt]{amsart} \ \usepackage{fullpage} \usepackage{psfig} \newcommand{\fr}[2]{\frac{#1}{#2}} \title[Problem Set 4]{ArsDigita University\\Month 2: Discrete Mathematics\\Professor Shai Simonson\\Problem Set 4 Answers} M \begin{document} \ \maketitle \ \begin{enumerate} \item {\bf Whats' wrong} \begin{enumerate} \item {\bf} The problem with the induction is that the base case is wrong. If the base case choosen is where n = 1 then at the base case you can have a binary number of either 1 or 0 thus disproving that it is the same. e \item {\bf} The problem with this induction is again the base case fails because you can not reach the number 22. n \end{enumerate} \ \item{\bf Second Problem} \begin{enumerate} \ \item {\bf Fibonacci Sequence}\\ \\ Assumption: $$ F(n) = (\fr{1}{\sqrt {5}})[(\fr{1}{2} + \fr{\sqrt{5}}{2}^n) - (\fr{1}{2} - \fr{\sqrt{5}}{2})^n]$$ Base Case: $$ F_1 = 1 \qquad F_0 = 0 $$ Prove: $$ F(n+1) = F(n) + F(n-1) \\$$ Let $a = (\fr{1}{2} + \fr{\sqrt{5}}{2})$\\ Let $b = (\fr{1}{2} - \fr{\sqrt{5}}{2})$\\ $$ \fr{1}{\sqrt{5}}(a^{n+1} - b^{n+1}) = \fr{1}{\sqrt{5}}(a^n - b^n) + \fr{1}{\sqrt{5}} (a^{n-1} - b^{n-1})) $$ \\ Divide both sides through by $\sqrt{5}$\\ $$a^{n+1} - b^{n+1} = a^n - b^n + a^{n-1} - b^{n-1}$$ \item {\bf The series}\\ \
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\ Assumption: $$1 + a + a^2 + a^3 \ldots + a^n = \fr{(1 - a^{n+1}}{1-a)}$$\\ Base Case: $$n = f(1)$$\\ $$= \fr{(1 - a^2)}{1 - a} = \fr{((1 - a)(1 + a))}{(1 - a)} = 1 + a$$\\ $$ n = 1 + a$$ Theory: $$\fr{(1 - a^{n+1})}{1-a + a^{n+1}} = \fr{(1 - a^{n+2})}{1 - a}$$ $$(1 - a)(1 - a^{n+1})/1-a + (1 -a)(a^{n+2}) = (1 - a)(1 - a^{n+2})/1 - a$$ $$(1 - a^{n+1}) + (1 - a)a^{n+1} = (1 - a^{n+2})$$ $$(1 - a^{n+1}) + a^{n+1} - a^{n+2} = (1 - a^{n+2})$$ $$1- a^{n+2} = (1 - a^{n+2})$$ qed. \\ \ \item {\bf The Mod Squad} \\ \ Assumption: $$4^{n+1} + 5^{2n-1} = 21a $$ where a is some constant $\geq $ 0\\ \\ Base Case: $$n = 1$$ $$4^{2} + 5^{1} = 21a$$ $$ 16 + 5 = 21a$$ $$ 21 = 21a$$ Theory: That for some n the assumption holds true.
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Problem_Set_04 - \documentclass[12pt]cfw_amsart...

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