% LESSON ONE
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\documentclass{amsart} 
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\begin{document}
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	\textbf{Lesson One: The Basics}
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\textbf{Font Families}

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I \textbf{love} math!

\LaTeX\ does \emph{not} rhyme with ``paychecks''!\\
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\textbf{Math Enviroments, Superscripts, and Subscripts}

The numbers 3, 4, and 5 are a Pythagorean triple because $3^2+4^2=5^2$.\\
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We all     know      that $(x^n)'=nx^{n-1}$, but what is $(x^x)'$?\\ 
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The recursive definition for the Fibonacci numbers is 

$F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n>2$.\\

Compare $\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$ to the following.

$$\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$$
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We can also display it midline by using ``display-style'': 
$\displaystyle\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$.\\
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\end{document}