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% Welcome to the MAT320 Homework template on Overleaf -- just edit your
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% Based on a homework template by Dana Ernst.
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% This is all preamble stuff that you don't have to worry about.
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\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amsthm,amssymb}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newenvironment{ex}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{sol}[1][Solution]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1:}]}{\end{trivlist}}
\begin{document}
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% Start here
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\noindent Firstname Lastname \hfill {\Large MAT320: Homework X} \hfill \today
\begin{ex}{3.2}
What is an example of a proof with mathematical content?
\end{ex}
\begin{sol}\
\begin{proof}
If you want to include math in a sentence, you use \$. For example (see latex file), $\int x = \frac{1}{2}x^2 + C$. If you want to display math (centered on a new line), use \$\$. For example (see latex file), $$\sum_{i=1}^{100} i = 5050.$$ Next is an example of the \texttt{align} environment:
%Note 1: The * tells LaTeX not to number the lines. If you remove the *, be sure to remove it below, too.
%Note 2: Inside the align environment, you do not want to use $-signs. The reason for this is that this is already a math environment. This is why we have to include \text{} around any text inside the align environment.
\begin{align*}
\sum_{i=1}^{k+1}i & = \left(\sum_{i=1}^{k}i\right) +(k+1)\\
& = \frac{k(k+1)}{2}+k+1 & (\text{by inductive hypothesis})\\
& = \frac{k(k+1)+2(k+1)}{2}\\
& = \frac{(k+1)(k+2)}{2}\\
& = \frac{(k+1)((k+1)+1)}{2}.
\end{align*}
\end{proof}
\end{sol}
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\end{document}
```