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\begin{document}

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\section*{Written Assignment 4}

\begin{question}{1}
Explain the difference between the ``Principle of Mathematical Induction'' and the ``\textit{Strong} Principle of Mathematical Induction.''
\end{question}

\begin{question}{2}
Prove by \underline{\textbf{induction}} that if $a$ is odd, then $a^n$ is odd, where $a, n \in \Z$ and $n \geq 1$.
\end{question}

\begin{question}{3}
Prove that $\forall~ n \geq 18$, there are integers $a$ and $b$ for which $n = 4a + 7b$; that is, $n$ is the sum of some number of 4s and 7s.
\end{question}

\begin{question}{4}
Solve book problem 3.6
\end{question}

\begin{question}{5}
Consider the following sequence defined recursively as:
% Left alignment of multiline equations
\begin{align*}
    g_1~&amp;=~ 1,&amp;\\
    g_n~&amp;=~g_{n-1}~+~2^{n-1}. ~~\forall n \geq 2
\end{align*}
\begin{enumerate}
    \item Compute:
    \begin{itemize}
        \item $g_2$
        \item $g_3$
        \item $g_4$
        \item $g_5$
    \end{itemize}
    \item Prove by mathematical induction that $g_{n} = 2^n - 1$.
\end{enumerate}
\end{question}

\end{document}
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