83 lines
2.0 KiB
TeX
83 lines
2.0 KiB
TeX
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[latin1]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {Mi 10-12 (Andre)}
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\def \uebung {1} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 1.1}
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\begin{enumerate}[(a)]
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\item
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\begin{itemize}
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\item Induktionsanfang:
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\begin{eqnarray*}
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H(n) = 2^n - 1 \\
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n = 0 \Rightarrow 2^0 - 1 = 0 \\
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n = 1 \Rightarrow 2^1 - 1 = 0 \\
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n = 2 \Rightarrow 2^2 - 1 = 0
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\end{eqnarray*}
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\item Induktionsvoraussetzung:
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$H(n) = 2^n - 1$ mit $n \in \mathbb{N}$
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\item Induktionsbehauptung:
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Zu zeigen, dass IV auf f\"ur $(n+1)$ gilt.
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\item Induktionsschritt:
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Algorithmus f\"ur $(n+1)$:
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\begin{itemize}
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\item $n$ Scheiben auf Hilfsstab legen ($2^n - 1$ Z\"uge)
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\item $n+1$-te Scheibe auf Zielstab legen ($1$ Zug)
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\item $n$ Scheiben von Hilfsstab auf Zielstab legen ($2^n - 1$ Z\"uge)
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\end{itemize}
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\begin{eqnarray*}
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H(n+1) &=& (2^n - 1) + 1 + (2^n - 1) \\
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&=& 2^n + 2^n - 1 \\
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&=& 2^{n+1} - 1
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\end{eqnarray*}
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\end{itemize}
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\item
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\begin{eqnarray*}
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\frac{2^{64} - 1}{60 \cdot 60 \cdot 24 \cdot 365} = 5,850 \cdot 10^{11} \approx 585\text{ Milliarden Jahre}
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\end{eqnarray*}
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\end{enumerate}
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\end{document}
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