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\documentclass{article}
\usepackage{amsmath,amsthm,booktabs}
\newtheorem{theorem}{Theorem}
\title{Sums of Odd Numbers}
\author{Erika Mustermann}
\date{November 2026}
\begin{document}
\maketitle
\begin{abstract}
We show that the first $n$ odd numbers add up to $n^2$.
\end{abstract}

\section{The claim}
Adding odd numbers one by one gives:
\begin{itemize}
  \item $1 = 1$,
  \item $1 + 3 = 4$,
  \item $1 + 3 + 5 = 9$.
\end{itemize}
The pattern suggests
\begin{equation}
  \sum_{k=1}^{n} (2k - 1) = n^2 . \label{eq:odd}
\end{equation}

\begin{theorem}
Equation~\eqref{eq:odd} holds for every $n \geq 1$.
\end{theorem}
\begin{proof}
For $n = 1$ both sides are 1. Adding the next odd
number $2n + 1$ to $n^2$ gives $(n+1)^2$.
\end{proof}

\section{A check}
\begin{tabular}{rr}
\toprule
$n$ & Sum \\
\midrule
4  & 16 \\
44 & 1936 \\
\bottomrule
\end{tabular}
\end{document}