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\title{General Mathematics and Computational Science I}
\author{Exercise 4}
\date{September 26, 2006}
\maketitle

\begin{enumerate}

\item (From Ivanov, p.\ 8) Prove that any sum of more than $7$ cents
can be made up out of $3$- and $5$-cent coins.

(Use the induction principle formulated in class!)

\item Using mathematical induction, prove that for all $n>1$ the
following inequality holds: \begin{equation}
\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{n^{2}}<1
\end{equation}
\textbf{Hint}: Prove that the sum on the left is less than or equal to
$1-\frac{1}{n}$.  (A similar problem can be found in Ivanov, p.~9,
Problem~7.)

\end{enumerate}
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