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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due September 2 (Tuesdsay)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\textbf{Topics}: Induction, the pigeonhole principle.\hfill\begin{tikzpicture}[baseline=4mm]\node at (0,0) {\reflectbox{\includegraphics[width=2.5cm]{pigeon2.jpg}}};\end{tikzpicture} \\
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\begin{enumerate}

\item Without the use of AI or Google, use induction to prove that if $a_n=1+2+\cdots +n$, then $a_n=n(n+1)/2$. Then ask ChatGPT if your solution is correct, and report what it says. You can get full credit for doing this problem, even if your original solution was flawed. \\ 

  
\item Without the use of AI or Googling a solution, prove that for all
  positive integers $n$, the equality
  \[
  1+2q+3q^2+\cdots+n q^{n-1}=\frac{1-q^{n+1}}{(1-q)^2}-(n+1)\frac{q^n}{1-q}
  \]
  holds for all complex numbers $q\neq 1$. [If you use AI or Google,
    please acknowledge it; you can still get full credit.] \\
  

\item Use ChatGPT to find surprising applications of the pigeonhole principal. In your own words, write up a complete solution. You will likely have to ask it multiple follow-up questions in order to get enough details to fully understand it. Include the prompts that you used. \\

  
\item Pick one of the exercises on Richard Stanley's worksheet on the pigenhole principle and solve it. This is from the Putnam Seminar at MIT, and you are encouraged to use ChatGPT for help. Include any prompts that you use. \\

  


\end{enumerate}

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