\documentclass{article}

\input{../../../preamble}

\begin{document}

\begin{theorem}[Sum of Arithmetic Series]

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

\end{theorem}

\begin{proof}

  \href{Sequence.lean}{Common.Data.Real.Sequence.Arithmetic.sum_recursive_closed}

\end{proof}

\begin{theorem}[Sum of Geometric Series]

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

\end{theorem}

\begin{proof}

  \href{Sequence.lean}{Common.Data.Real.Sequence.Geometric.sum_recursive_closed}

\end{proof}

\end{document}
