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\title{\sc Math 2100 / 2105 / 2350 -- Homework 4}
\author{due Wednesday, March 27, at the beginning of class}
\date{}
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\begin{center} 
	\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}

\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. \textbf{Staple your work} together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive credit. Explain all reasoning.

\begin{enumerate}
	\item Prove that if $3$ divides $4^{n-1}-1$ then $3$ divides $4^n-1$.
	\item Prove that for all positive integers $n \geq 4$,
	\[
		n! > 2^n.
	\]
	\item Prove  that for all positive integers $n$, the number $5^{2n+1} + 2^{2n+1}$ is divisible by $7$.
	\item Prove that for all positive \textbf{odd} integers $m$, the number $m^2-1$ is divisible by $8$.
	\item Prove that for all positive integers $n$,
	\[
		1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2.
	\]
	You may use the theorem we proved from class that says
	\[
		1 + 2 + \cdots + n = \frac{n(n+1)}{2}.
	\]
	\item Prove that for all positive integers $n$,
	\[
		\sum_{k=0}^n (k \cdot k!) = (n+1)!-1.
	\]
	\item Prove that for all $n \geq 0$:
	\[
		\sum_{i=0}^n i2^i = 2 + (n-1)2^{n+1}.
	\]
	\item Prove that for all $n \geq 1$:
	\[
		1 + 3 + 5 + \cdots + (2n-1) = n^2.
	\]


	
\end{enumerate}
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