36 lines
765 B
TeX
36 lines
765 B
TeX
\documentclass{article}
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\input{../../../preamble}
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\begin{document}
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\begin{theorem}[Sum of Arithmetic Series]
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence.lean}{Common.Data.Real.Sequence.Arithmetic.sum_recursive_closed}
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\end{proof}
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\begin{theorem}[Sum of Geometric Series]
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Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence.lean}{Common.Data.Real.Sequence.Geometric.sum_recursive_closed}
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\end{proof}
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\end{document}
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