\documentclass[12pt]{article}
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\usepackage{times,mathptm}
\usepackage{amsmath,times,mathptm}

\input{ExamMacros.tex}


\prof{Chris Bendel and Peter Cholak}
\class{Math 222}
\quizdate{Wednesday January 20}
\pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\

%\begin{problem}
%  Show that for any positive integer $n$ a set with exactly $n$
%  elements contains exactly $2^n$ subsets.  (Counting the empty set
%  and the whole set.)  {\bf Hint:} Use induction.
%\end{problem}

\begin{problem}
  In the ``real'' world, we are used to the fact that a product of two
  non-zero numbers is again non-zero.  We will later see that this
  fact is not true in all worlds.  Show that the complex numbers do
  behave normally.  Precisely, Let $z$ and $w$ be two arbitrary
  complex numbers and suppose that the product $zw = 0$.  Show that at
  least one of $z$ or $w$ must be zero.
\end{problem}

%\vspace{3in}

%\begin{problem}
%  Show by induction that $n^4 -4n^2$ is divisible by $3$ for all $n
%  \geq 1$.
%\end{problem}

\begin{problem}
  Show $ (1) (2) ( 3) + (2)(3)(4) + (3)(4)( 5) + \ldots + n (n+1)
  (n+2) = \frac{n(n+1)(n+2)(n+3)}{4}$.
\end{problem}

\begin{problem}
  The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots.  In
  general, the Fibonacci numbers are defined by $f_1 = 1$, $f_2 = 1$,
  and $f_{n+2} = f_{n+1} + f_{n}$ for $n = 1,2, 3, \dots$.  Prove that
  the $n$th Fibonacci number $f_n$ satisfies $f_n < 2^n$.  {\bf Hint:}
  Use the ``second'' version of induction and you will have to prove
  {\em two} starting cases.
\end{problem}

\end{document}
             