\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{url}
\usepackage{amsmath}

\title{Activity 2.1}
\author{Jesús Carro}
\date{March 2021}

\begin{document}

\maketitle

\section{First part}
In-line mode: $a^2 = b^2 + c^2$. Display mode: $$a^2=b^2+c^2$$
Now, a numbered equation:
\begin{equation}
    a^2=b^2+c^2
\end{equation}
All is in the same paragraph for \LaTeX{}.

\section{Second part}

\paragraph{Formula 1}
$$\int_0^1 f(x)\ dx$$

\paragraph{Formula 2}
\begin{equation*}
A=\left(\begin{matrix}
        1 & 2 & 3\\
        0 & 7 & c\\
        a & -1 & b
    \end{matrix}\right) \implies 
    |A|=\left|\begin{matrix}
        1 & 2 & 3\\
        0 & 7 & c\\
        a & -1 & b
    \end{matrix}\right|
\end{equation*}

\paragraph{Formula 3}

\begin{equation*}
    \sum_{n=0}^\infty \frac{1+n^2}{n^3}
\end{equation*}

\paragraph{Formula 4}

\begin{equation*}
    \prod_{k=1}^\infty\left(\sum_{n=0}^k \frac{1+n^2}{n^3}\right)
\end{equation*}

\paragraph{Formula 5}

\begin{equation*}
    \binom{n}{k} = \frac{n!}{k!(n-k)!} 
\end{equation*}


\paragraph{Formula 6 (Extra, with other character for matrices)}

\begin{equation*}
    A_{M\times N} = \left(\begin{matrix}
    a_{1,1} & a_{1,2} & \hdots & a_{1,N}\\
    a_{2,1} & a_{2,2} & \hdots & a_{2,N}\\
    \vdots & \vdots & \ddots & \vdots\\
    a_{M,1} & a_{M,2} & \hdots & a_{M,N}\\
    \end{matrix}\right)
\end{equation*}

\paragraph{Formula 7 (Extra, with Greek characters)}

\begin{equation*}
    \frac{df(t)}{dt} = \lim_{\delta \to 0} \frac{f(t+\delta)-f(t)}{\delta}
\end{equation*}

\paragraph{Formula 8 (Extra, with Greek characters)}

\begin{equation*}
    f(x) = \left\{\begin{matrix}
    2-\frac{\tan(x)}{x} & - \frac{\pi}{2} < x < 0\\
    1 & x = 0\\
    \frac{\tan(x)}{x} & 0 < x < \frac{\pi}{2}
    \end{matrix}\right.
\end{equation*}

\section{Tupper's self-referential formula (optional)}

$$\frac{1}{2}<\left\lfloor mod\left(\left\lfloor \frac{y}{17}^{-17\lfloor x \rfloor - mod(\lfloor y \rfloor, 17)}\right\rfloor , 2\right)\right\rfloor$$
\end{document}
