\documentclass{article}

\input{../../preamble}

\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}

\begin{document}

\header{Sequences}{}

\tableofcontents

\section{Summations}%
\label{sec:summations}

\subsection{\unverified{Arithmetic Series}}%
\label{sub:sum-arithmetic-series}

Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
  Then for some $n \in \mathbb{N}$,
  $$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$

\begin{proof}

  \lean{Common/Real/Sequence/Arithmetic}
    {Real.Arithmetic.sum\_recursive\_closed}

\end{proof}

\subsection{\unverified{Geometric Series}}%
\label{sub:sum-geometric-series}

Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
  Then for some $n \in \mathbb{N}$,
  $$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$

\begin{proof}

  \lean{Common/Real/Sequence/Geometric}
    {Real.Geometric.sum\_recursive\_closed}

\end{proof}

\end{document}