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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Foundations \hspace{.7in} Due 1/30/17} \end{center}

\noindent Four of these problems will be graded (our choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item $\sqrt{5}$ is irrational.
\item The sum of two rational numbers is rational.
\item The sum of two irrational numbers is irrational.
\item The product of a rational and an irrational is irrational.
\item Consider the formula $1+2+3+...+ n = \frac{n^2+n+1}{2}$.
\begin{enumerate}
	\item Write the formula in sigma notation.
	\item Show that if this formula works for $n = k$, then it also must work for $n = k + 1$.
	\item Explain why mathematical induction does not prove that this formula is true for all $n \in \mathbb N$.
\end{enumerate}
\item For any $n \in \mathbb N$, with $n \ge 1$, $$\sum_{i=1}^{n} c = nc$$
\item For any $n \in \mathbb N$, with $n \ge 1$, $$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
\item For any integer $n$, the number $n^2 + n + 17$ is prime.


\end{enumerate}

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