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2024-04-25 16:39:23 -07:00
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7
Misc/Proof Techniques/parts/2 induction.tex
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@ -189,11 +189,8 @@ For example, see the proof of the statement in \ref{binomsum} on the next page.
\pagebreak
\makeatletter
\@makeORMCbox{tmpbox}{Alternative Proof}{ogrape!10!white}{ogrape}
\makeatother
\begin{tmpbox}
\begin{ORMCbox}{Alternative Proof}{ogrape!10!white}{ogrape}
Consider the following problem: \par
How many ways are there to write a number $x$ as an ordered sum of positive integers? \par
\note{
@ -237,7 +234,7 @@ For example, see the proof of the statement in \ref{binomsum} on the next page.
We've found that the number of ways to split $x$ can be written as either
$\sum_{n = 1}^{x-1} \binom{x-1}{n}$ or $2^{x-1}$,
and therefore $\sum_{n = 1}^{x-1} \binom{x-1}{n} = 2^{x-1}$.
\end{tmpbox}
\end{ORMCbox}
\pagebreak