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fix(proofs): reformat equation to be multiline

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Barrett Ruth 2025-11-09 15:07:46 -05:00
parent 0aaf0f13c8
commit 1de9c3decb
1 changed files with 3 additions and 3 deletions
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6
src/content/algorithms/proofs.mdx
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@ -83,9 +83,9 @@ As $S_i<S_{j>i}$, $|S_{j>i}-S_i|=S_{j>i}-S_i$--the sum is unchanged. Rewriting t
$$
\begin{align*}
\sum_{1\leq r\leq n}\sum_{1\leq l<r} |X-Y| &=\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_r-\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_l
&=\sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq l<r\leq n} S_l
&=\sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq r\leq n} (n-r)\cdot S_r
\sum_{1\leq r\leq n}\sum_{1\leq l<r} |X-Y| &= \sum_{1\leq r\leq n}\sum_{1\leq l<r}S_r-\sum_{1\leq r\leq n}\sum_{1\leq l<r}S_l \\
&= \sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq l<r\leq n} S_l \\
&= \sum_{1\leq r\leq n}r\cdot S_r-\sum_{1\leq r\leq n} (n-r)\cdot S_r
\end{align*}
$$