# Proving <munderover> &#x220F;<!-- ∏ --> <mrow class="MJX-TeXAtom-ORD"> i =

Proving $\prod _{i=1}^{n}\left(2i-1\right)$ for all natural numbers
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We have,
$\prod _{\phantom{\rule{thinmathspace}{0ex}}i=1}^{\phantom{\rule{thinmathspace}{0ex}}n}\left(2i-1\right)$
$=1\cdot 3\cdot 5\cdot 7\cdots \left(2n-1\right)$
$=\frac{1\cdot 3\cdot 5\cdot 7\cdots \left(2n-1\right)\cdot 2\cdot 4\cdot 6\cdots 2n}{2\cdot 4\cdot 6\cdots 2n}$
$=\frac{1\cdot 2\cdot 3\cdot 4\cdots \phantom{\rule{thinmathspace}{0ex}}2n}{{2}^{n}\left(1\cdot 2\cdot 3\cdots \phantom{\rule{thinmathspace}{0ex}}n\right)}$
$=\frac{\left(2n\right)!}{{2}^{n}\cdot \phantom{\rule{thinmathspace}{0ex}}n!}$