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\begin{document}

\begin{center}
{\large{\textsc{Homework 8}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item In class, we saw a \textbf{correct} proof of the following statement: For each $n \in \NN$,
\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]

Consider the \textbf{incorrect} proof of the same statement given below.

\noindent\fbox{
\parbox{5.5in}{
\begin{proof}
\textbf{Base Case:} Since $1 = \frac{1 \cdot 2}{2}$, $P(1)$ is true. \\

\textbf{Inductive Step:} Let $n \in \NN$. Then
\begin{align*}
1 + 2 + \cdots + n + (n+1) &= \frac{(n+1)(n+2)}{2} \\
\frac{n(n+1)}{2} + (n+1) &= \frac{(n+1)(n+2)}{2} \\
n+1 &= \frac{(n+1)(n+2)}{2} - \frac{n(n+1)}{2} \\
n+1 &= \bigg(\frac{n+1}{2}\bigg) \big( (n+2) - n \big) \\
n+1 &= n+1.
\end{align*}
Thus, $P(n+1)$ is true.
\end{proof}
}
}\\

Explain, in complete sentences, what is wrong with this proof. How would you fix it?

\vspace{.1in}

\item \textbf{{[Falkner Section 4 Exercise 1]}} Let $x$, $y$, and $z$ be integers.
\begin{enumerate}
\item Prove that if $x$ is even and $y$ is even, then $x+y$ is even.
\item Prove that if $x$ is even and $y$ is odd, then $x+y$ is odd.
\item Suppose $x$, $y$, and $z$ are odd. Is $x + y + z$ odd? Or is $x + y + z$ even? Explain your answer. You should not have to use the definitions of odd and even. Instead, you should be able to answer this part by combining one of parts (a) and (b) with Example 4.5.
\end{enumerate}

\newpage

\item \textbf{{[Falkner Section 4 Exercise 2]}} Let $x$, $y$, and $z$ be integers.
\begin{enumerate}
\item Prove that if $x$ is odd and $y$ is odd, then $xy$ is odd.
\item Suppose $x$, $y$, and $z$ are odd. Is $xyz$ odd? Or is $xyz$ even? You should not have to use the definitions of odd and even. Instead, you should be able to answer this part by applying part (a).
\end{enumerate}

\end{enumerate}


\vspace{.25 in}


\begin{center}
\textbf{Practice Problems}
\end{center}

It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Falkner Section 4 Exercise 3]}} Let $x$ be an integer. Prove that $x(x+1)$ is even.

\item Let $r \neq 1$ be a real number. Prove that for every $n \in \NN$,
\[ \displaystyle 1+r+r^2+\cdots +r^{n-1}=\frac{r^n-1}{r-1}. \]
This is the (finite) \textbf{geometric series formula}.
\end{enumerate}




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