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\begin{document}

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\begin{center}
\textbf{\large Mathematik für Wirtschaftsinformatik}
\end{center}
\textbf{Wintersemester 2015/2016\hfill Prof. Dr. M. Keller}

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\textbf{Blatt 7}\hfill % Nr des Blatts
\textbf{Abgabe 17.12.2015}

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\begin{itemize}
  \item[(1)] Zeigen Sie, dass für alle $n\in\N$ die Gleichung \[\sum_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{6}\] gilt. 
  
\item[(2)] Zeigen Sie, dass für  alle $n\in\N$ die Zahl $3^{2n+1}-3$ durch $6$ teilbar ist.

\item[(3)] Es sei $f:\N\to\IQ$ definiert durch $f(1)=2$ und $f(n+1)=2-\frac{1}{f(n)}$ für $n>1$. Zeigen Sie, dass für alle $n\in\N$ die Gleichung 
\[f(n)=\frac{n+1}{n}\] gilt.

\item[(4)] (Bernoulli Ungleichung) Zeigen Sie, dass für alle $x\geq-1$ und alle $n\in\IN$ die Ungleichung
\[(1+x)^n\geq 1+nx\] gilt.\\
Tipp: Es ist $(1+x)^{n+1}=(1+x)^n(1+x)$ und es gilt $x^2\geq 0$.

\end{itemize}
\end{document}