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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 10 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

%%------------------------------------------------------------------------

\begin{exercise} % HW 4, Problem 1
Prove that for all integers $n\geq 1$,
\[
1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}.
\]
\end{exercise}

%\begin{solution} 
    %% Reminder to cite AI usage, if applicable. 
    % Put it immediately after \begin{solution} (on the same line)
    %\AIpolish
    %\AImajor
    %\AIminor
    %\AIpolishmajor
    %\AIpolishminor
    %
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 4, Problem 2
Prove the following inequality for all integers $n\geq2$:
\[
\sqrt{n}<1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{n}}.
\]
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 4, Problem 3
Recall that the Fibonacci numbers are defined by the recurrence $F_{n+2}=F_{n+1}+F_n$, with $F_1=1$ and $F_2=1$. Prove that 
\[
F_n\leq 2^{n-1}\qquad\text{for all $n\geq 1$}.
\]
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\smallskip 

%%------------------------------------------------------------------------

\begin{exercise} % HW 4, Problem 4
Prove that for any $n+1$ distinct numbers, there will be two whose difference is divisible by $n$.
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 4, Problem 5
Construct truth tables that demonstrate the following tautologies, and include the intermediate steps as columns.
\begin{enumerate}[label=(\roman*)]
    \item \emph{Modus tollens}:  $\big[\neg Q\;\And\;(P\Rightarrow Q)\big]\,\Rightarrow\,\neg P$
    \item Contradiction:  $P\Leftrightarrow[(\neg P)\Rightarrow (Q\And\neg Q)]$
\end{enumerate}
Rather than include this as an exercise in your manuscript, weave it into your narrative directly in the chapter, right after \emph{modus ponens}.
\end{exercise}

%%------------------------------------------------------------------------

\end{document}
