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\title{\sc Math 2100 -- Homework 4}
\author{
Fall 2024\\[10pt]
due Wednesday, \textbf{October 30}, \underline{at the beginning of class}\\[10pt]
Sections 2.3, 2.4, some 2.5
}
\date{}
\begin{document}
\maketitle
\begin{center}
\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}
\textbf{{\Large $\star$} It is not permitted to use any AI tools or Large Language Models (ChatGPT, Claude, Gemini, etc) to assist with this assignment. {\Large $\star$}}
\textbf{Please read the syllabus to remind yourself of our \underline{collaboration policy}.}
\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. It may be handwritten (as long as I can read it) or typed with software such as Word or Latex. Please write the questions in the correct order. Explain all reasoning.
\hrulefill
\begin{enumerate}
\item Prove that for all positive integers $n$,
\[
\ds\sum_{k=0}^n (k \cdot k!) = (n+1)! - 1.
\]
\item Prove that $\ds\sum_{i=1}^n \frac{1}{(i)(i+1)} = \frac{n}{n+1}$ for all $n \geq 1$.
\item Prove that for all $n \in \N$, the number $9^n - 1$ is divisible by $8$.
\item Prove that for all positive integers $n \geq 4$,
\[
n! > 2^n.
\]
\item Use induction to prove that for all integers $n \geq 0$, the quantity $2^{2n+1} + 5^{2n+1}$ is divisible by $7$.
\item Prove that $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$.
\item Prove that at a completely full Milwaukee Bucks game at the Fiserv Forum, there \emph{must} be at least two people that have both the same birthday \emph{and} the same first initial. (Note: you will have to look up the capacity of the arena!)
\item Use the pigeonhole principle to prove that given any five integers, there will be two that have a sum or difference divisible by 7.
\item Prove that if any five points other than $(0,0)$ are placed on the coordinate plane, then there are two points, call them $A$ and $B$, such that the angle formed by the rays from $(0,0)$ to $A$ and from $(0,0)$ to $B$ is acute.
\end{enumerate}
\end{document}