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\title{Math 2001 \\ Proof by induction}

\begin{document}
\maketitle



\begin{quotation}
\textbf{General outline of a proof by induction.}
\begin{proof}\
\begin{enumerate}[\qquad 1.]
\item Base case: verify that the first statement is true.
\item Induction step: show that if the $k$-th statement is true, then the $(k+1)$-st statement is true.
\end{enumerate}
\end{proof}
\end{quotation}



\begin{example}
Prove that $7 \mid 4^{3n}-1$ for every non-negative integer $n$ (i.e. $n = 0, 1, 2, 3, \dotsc$).
\begin{proof}
{\color{white}.}

\vspace{60mm}
\end{proof}
\end{example}

\begin{example}
Prove that $\ds\sum_{i=1}^n i= \frac{n(n+1)}{2}$.
\begin{proof}
{\color{white}.}

\vspace{60mm}
\end{proof}
\end{example}

\pagebreak

\begin{example}
Let $a_1, a_2, a_3, \dotsc$ be a sequence of integers where $a_1 = 3$, $a_2 = 1$, and $a_n = a_{n-2}+a_{n-1}$ for each integer $n \ge 4$. Prove that $1 \le \frac{a_n}{a_{n-1}} \le 2$ for each $n \ge 4$.
\begin{proof}
{\color{white}.}

\vspace{80mm}
\end{proof}
\end{example}

\begin{example}
Let $A_1, A_2, \dotsc, A_n$ be sets. Prove that $\overline{A_1 \cap A_2 \cap \dotsb \cap A_n} = \overline{A_1} \cup \overline{A_2} \cup \dotsc \cup \overline{A_n}$.
\begin{proof}
{\color{white}.}

\vfill
\end{proof}
\end{example}


\textbf{Suggested proofs:} 

Chapter 10 (induction): 6, 7, 18. 

Chapter 12.6 (functions and inverses): 10, 11, 13.


\end{document}